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Section5.2Substitution

In Section 5.1, we focused on solving systems of equations by graphing. In addition to being time consuming, graphing can be an awkward method to determine the exact solution when the solution has large numbers or fractions. There are two symbolic methods for solving systems of linear equations, and in this section we will use one of them: substitution.

Figure5.2.1Alternative Video Lesson

Subsection5.2.1Solving Systems of Equations Using Substitution

Example5.2.2The Interview

In 2014, the New York Times 1 (nyti.ms/2pupebT) posted the following about the movie, “The Interview”:

“The Interview” generated roughly \(\$15\) million in online sales and rentals during its first four days of availability, Sony Pictures said on Sunday.

Sony did not say how much of that total represented \(\$6\) digital rentals versus \(\$15\) sales. The studio said there were about two million transactions overall.

A few days later, Joey Devilla cleverly pointed out in his blog 2 (http://www.joeydevilla.com/2014/12/31/), that there is enough information given to find the amount of sales versus rentals. Using algebra, we can write a system of equations and solve it to find the two quantities. 3 Although since the given information uses approximate values, the solutions we will find will only be approximations too.

First, we will define variables. We need two variables, because there are two unknown quantities: how many sales there were and how many rentals there were. Let \(r\) be the number of rental transactions and let \(s\) be the number of sales transactions.

If you are unsure how to write an equation from the background information, use the units to help you. The units of each term in an equation must match because we can only add like quantities. Both \(r\) and \(s\) are in transactions. The article says that the total number of transactions is \(2\) million. So our first equation will add the total number of rental and sales transactions and set that equal to \(2\) million. Our equation is:

\begin{equation*} (r\,\text{transactions})+(s\,\text{transactions})=2{,}000{,}000\,\text{transactions} \end{equation*}

Without the units:

\begin{equation*} r+s=2{,}000{,}000 \end{equation*}

The price of each rental was \(\$6\text{.}\) That means the problem has given us a rate of \(6\,\frac{\text{dollars}}{\text{transaction}}\) to work with. The rate unit suggests this should be multiplied by something measured in transactions. It makes sense to multiply by \(r\text{,}\) and then the number of dollars generated from rentals was \(6r\text{.}\) Similarly, the price of each sale was \(\$15\text{,}\) so the revenue from sales was \(15s\text{.}\) The total revenue was \(\$15\) million, which we can represent with this equation:

\begin{equation*} \left(6\,\tfrac{\text{dollars}}{\text{transaction}}\right)(r\,\text{transactions})+\left(15\,\tfrac{\text{dollars}}{\text{transaction}}\right)(s\,\text{transactions})=\$15{,}000{,}000 \end{equation*}

Without the units:

\begin{equation*} 6r+15s=15{,}000{,}000 \end{equation*}

Here is our system of equations:

\begin{equation*} \left\{ \begin{alignedat}{4} r\amp+{}\amp s\amp={}\amp2{,}000{,}000 \\ 6r\amp+{}\amp 15s\amp={}\amp15{,}000{,}000 \end{alignedat} \right. \end{equation*}

To solve the system, we will use the substitution method. The idea is to use one equation to find an expression that is equal to \(r\) but, cleverly, does not use the variable “\(r\)”. Then, substitute this for \(r\) into the other equation. This leaves you with one equation that only has one variable.

The first equation from the system is an easy one to solve for \(r\text{:}\)

\begin{align*} r+s \amp=2{,}000{,}000 \\ r+s \subtractright{s} \amp=2{,}000{,}000\subtractright{s} \\ r \amp=2{,}000{,}000-s \end{align*}

This tells us that the expression \(2{,}000{,}000-s\) is equal to \(r\text{,}\) so we can substitute it for \(r\) in the second equation:

\begin{align*} 6r+15s \amp=15{,}000{,}000\\ 6(\substitute{2{,}000{,}000-s})+15s \amp=15{,}000{,}000\\ \end{align*}

Now we have an equation with only one variable, \(s\text{,}\) which we will solve for:

\begin{align*} 6(2{,}000{,}000-s)+15s \amp=15{,}000{,}000\\ 12{,}000{,}000-6s+15s \amp=15{,}000{,}000\\ 12{,}000{,}000+9s \amp= 15{,}000{,}000\\ 12{,}000{,}000+9s\subtractright{12{,}000{,}000} \amp= 15{,}000{,}000\subtractright{12{,}000{,}000}\\ 9s \amp= 3{,}000{,}000\\ \divideunder{9s}{9} \amp= \divideunder{3{,}000{,}000}{9}\\ s \amp= 333{,}333.\overline{3} \end{align*}

At this point, we know that \(s=333{,}333.\overline{3}\text{.}\) This tells us that out of the \(2\) million transactions, roughly \(333{,}333\) were from online sales. Recall that we solved the first equation for \(r\text{,}\) and found \(r=2{,}000{,}000-s\text{.}\)

\begin{align*} r \amp=2{,}000{,}000-s\\ r \amp=2{,}000{,}000-\substitute{333{,}333.\overline{3}}\\ r \amp=1{,}666{,}666.\overline{6} \end{align*}

To check our answer, we will see if \(s=333{,}333.\overline{3}\) and \(r=1{,}666{,}666.\overline{6}\) make the original equations true:

\begin{align*} r+s \amp=2{,}000{,}000\\ \substitute{1{,}666{,}666.\overline{6}}+\substitute{333{,}333.\overline{3}} \amp\stackrel{?}{=}2{,}000{,}000\\ 2{,}000{,}000\amp\stackrel{\checkmark}{=}2{,}000{,}000 \end{align*} \begin{align*} 6r+15s \amp=15{,}000{,}000\\ 6\left(\substitute{1{,}666{,}666.\overline{6}}\right)+15\left(\substitute{333{,}333.\overline{3}}\right) \amp\stackrel{?}{=}15{,}000{,}000\\ 10{,}000{,}000+5{,}000{,}000 \amp\stackrel{\checkmark}{=}15{,}000{,}000 \end{align*}

In summary, there were roughly \(333{,}333\) copies sold and roughly \(1{,}666{,}667\) copies rented.

Remark5.2.3

In Example 5.2.2, we chose to solve the equation \(r+s=2{,}000{,}000\) for \(r\text{.}\) We could just as easily have instead solved for \(s\) and substituted that result into the second equation instead. The summary conclusion would have been the same.

Remark5.2.4

In Example 5.2.2, we rounded the solution values because only whole numbers make sense in the context of the problem. It was OK to round, because the original information we had to work with were rounded. In fact, it would be OK to round even more to \(s=330{,}000\) and \(r=1{,}700{,}000\text{,}\) as long as we communicate clearly that we rounded and our values are rough.

In other exercises where there is no context and nothing suggests the given numbers are approximations, it is not OK to round and all answers should be communicated with their exact values.

Example5.2.5

Solve the system of equations using substitution:

\begin{align*} \left\{ \begin{alignedat}{4} x\amp+{}2y \amp={} 8 \\ 3x\amp-{}2y \amp={} 8 \\ \end{alignedat} \right. \end{align*}
Solution

To use substitution, we need to solve for one of the variables in one of our equations. Looking at both equations, it will be easiest to solve for \(x\) in the first equation:

\begin{align*} x+2y\amp=8\\ x+2y\subtractright{2y}\amp=8\subtractright{2y}\\ x\amp= 8-2y \end{align*}

Next, we replace \(x\) in the second equation with \(8-2y\text{,}\) giving us a linear equation in only one variable, \(y\text{,}\) that we may solve:

\begin{align*} 3x-2y\amp=8\\ 3(\substitute{8-2y})-2y\amp=8\\ 24-6y-2y\amp=8\\ 24-8y\amp=8\\ 24-8y\subtractright{24}\amp=8\subtractright{24}\\ -8y\amp=-16\\ \divideunder{-8y}{-8}\amp=\divideunder{-16}{-8}\\ y\amp=2 \end{align*}

Now that we have the value for \(y\text{,}\) we need to find the value for \(x\text{.}\) We have already solved the first equation for \(x\text{,}\) so that is the easiest equation to use.

\begin{align*} x\amp= 8-2y\\ x\amp= 8-2(\substitute{2})\\ x\amp= 8-4\\ x\amp=4 \end{align*}

To check this solution, we replace \(x\) with \(4\) and \(y\) with \(2\) in each equation:

\begin{align*} x+2y\amp=8\amp3x-2y\amp=8\\ \substitute{4}+2\substitute{(2)}\amp\stackrel{?}{=}8\amp3\substitute{(4)}-2\substitute{(2)}\amp\stackrel{?}{=}8\\ 4+4\amp\stackrel{\checkmark}{=}8\amp12-4\amp\stackrel{\checkmark}{=}8 \end{align*}

We conclude then that this system of equations is true when \(x=4\) and \(y=2\text{.}\) Our solution is the point \((4,2)\) and we write the solution set as \(\{(4,2)\}\text{.}\)

Exercise5.2.6

Try a similar exercise.

Example5.2.7

Solve this system of equations using substitution:

\begin{align*} \left\{ \begin{alignedat}{4} 3x\amp-{}7y \amp={} 5 \\ -5x\amp+{}2y \amp={} 11 \\ \end{alignedat} \right. \end{align*}
Solution

We need to solve for one of the variables in one of our equations. Looking at both equations, it will be easiest to solve for \(y\) in the second equation. The coefficient of \(y\) in that equation is smallest.

\begin{align*} -5x+2y\amp=11\\ -5x+2y\addright{5x}\amp=11\addright{5x}\\ 2y\amp=11+5x\\ \divideunder{2y}{2}\amp=\divideunder{11+5x}{2}\\ y\amp=\frac{11}{2}+\frac{5}{2}x \end{align*}

Note that in this example, there are fractions once we solve for \(y\text{.}\) We should take care with the steps that follow that the fraction arithmetic is correct.

Replace \(y\) in the first equation with \(\frac{11}{2}+\frac{5}{2}x\text{,}\) giving us a linear equation in only one variable, \(x\text{,}\) that we may solve:

\begin{align*} 3x-7y\amp=5\\ 3x-7\left(\frac{11}{2}+\frac{5}{2}x\right)\amp=5\\ 3x-7\cdot\frac{11}{2}-7\cdot\frac{5}{2}x\amp=5\\ 3x-\frac{77}{2}-\frac{35}{2}x\amp=5\\ \frac{6}{2}x-\frac{77}{2}-\frac{35}{2}x\amp=5\\ -\frac{29}{2}x-\frac{77}{2}\amp=5\\ -\frac{29}{2}x-\frac{77}{2}\addright{\frac{77}{2}}\amp=5\addright{\frac{77}{2}}\\ -\frac{29}{2}x\amp=\frac{10}{2}\addright{\frac{77}{2}}\\ -\frac{29}{2}x\amp=\frac{87}{2}\\ \multiplyleft{-\frac{2}{29}}\left(-\frac{29}{2}x\right)\amp=\multiplyleft{-\frac{2}{29}}\left(\frac{87}{2}\right)\\ x\amp=-3 \end{align*}

Now that we have the value for \(x\text{,}\) we need to find the value for \(y\text{.}\) We have already solved the second equation for \(y\text{,}\) so that is the easiest equation to use.

\begin{align*} y\amp=\frac{11}{2}+\frac{5}{2}x\\ y\amp=\frac{11}{2}+\frac{5}{2}(\substitute{-3})\\ y\amp=\frac{11}{2}-\frac{15}{2}\\ y\amp=-\frac{4}{2}\\ y\amp=-2 \end{align*}

To check this solution, we replace \(x\) with \(-3\) and \(y\) with \(-2\) in each equation:

\begin{align*} 3x-7y \amp=5 \amp -5x+2y \amp= 11\\ 3(-3)-7(-2) \amp\stackrel{?}{=}5 \amp -5(-3)+2(-2) \amp\stackrel{?}{=} 11\\ -9+14\amp\stackrel{\checkmark}{=}5\amp 15-4\amp\stackrel{\checkmark}{=}11 \end{align*}

We conclude then that this system of equations is true when \(x=-3\) and \(y=-2\text{.}\) Our solution is the point \((-3,-2)\) and we write the solution set as \(\{(-3,-2)\}\text{.}\)

Example5.2.8Clearing Fraction Denominators Before Solving

Solve the system of equations using the substitution method:

\begin{align*} \left\{ \begin{aligned} \frac{x}{3} - \frac{1}{2}y \amp= \frac{5}{6} \\ \frac{1}{4}x \amp = \frac{y}{2} + 1 \\ \end{aligned} \right. \end{align*}
Solution

When a system of equations has fraction coefficients, it can be helpful to take steps that replace the fractions with whole numbers. With each equation, we may multiply each side by the least common multiple of all the denominators.

In the first equation, the least common multiple of the denominators is \(6\text{,}\) so:

\begin{align*} \frac{x}{3}-\frac{1}{2}y\amp=\frac{5}{6}\\ \multiplyleft{6}\left(\frac{x}{3}-\frac{1}{2}y\right)\amp=\multiplyleft{6}\frac{5}{6}\\ 6\cdot\frac{x}{3}-6\cdot\frac{1}{2}y\amp=\multiplyleft{6}\frac{5}{6}\\ 2x-3y\amp=5 \end{align*}

In the second equation, the least common multiple of the denominators is \(4\text{,}\) so:

\begin{align*} \frac{1}{4}x\amp=\frac{y}{2}+1\\ \multiplyleft{4}\frac{1}{4}x\amp=\multiplyleft{4}\frac{y}{2}+\multiplyleft{4}1\\ \multiplyleft{4}\frac{1}{4}x\amp=\multiplyleft{4}\frac{y}{2}+\multiplyleft{4}1\\ x\amp=2y+4 \end{align*}

Now we have this system that is equivalent to the original system of equations, but there are no fraction coefficients:

\begin{align*} \left\{ \begin{aligned} 2x-3y\amp=5 \\ x\amp=2y+4 \\ \end{aligned} \right. \end{align*}

The second equation is already solved for \(x\text{,}\) so we will substitute \(x\) in the first equation with \(2y+4\text{,}\) and we have:

\begin{align*} 2x-3y\amp=5\\ 2(\substitute{2y+4})-3y\amp=5\\ 4y+8-3y\amp=5\\ y+8\amp=5\\ y\amp=-3 \end{align*}

And we have solved for \(y\text{.}\) To find \(x\text{,}\) we know \(x=2y+4\text{,}\) so we have:

\begin{align*} x\amp=2y+4\\ x\amp=2(\substitute{-3})+4\\ x\amp=-6+4\\ x\amp=-2 \end{align*}

The solution is \((-2,-3)\text{.}\) Checking this solution is left as an exercise.

Exercise5.2.9

Try a similar exercise.

Subsection5.2.2Applications of Systems of Equations

In Example 5.2.2, we set up and solved a system of linear equations for a real-world application. The quantities in that problem included rate units (dollars per transaction). Here are some more scenarios that we can model with systems of linear equations.

Example5.2.10Two Different Interest Rates

Greta made some large purchases with her two credit cards one month and took on a total of \(\$8{,}400\) in debt from the two cards. She didn't make any payments the first month, so the two credit card debts each started to accrue interest. That month, her Visa card charged \(2\%\) interest and her Mastercard charged \(2.5\%\) interest. Because of this, Greta's total debt grew by \(\$178\text{.}\) How much money did Greta charge to each card?

Solution

To start, we will define two variables based on our two unknowns. Let \(v\) be the amount charged to the Visa card (in dollars) and let \(m\) be the amount charged to the Mastercard (in dollars).

To determine our equations, notice that we are given two different totals. We will use these to form our two equations. The total amount charged is \(\$8{,}400\) so we have:

\begin{equation*} (v\,\text{dollars})+(m\,\text{dollars})=\$8400 \end{equation*}

Or without units:

\begin{equation*} v+m=8400 \end{equation*}

The other total we were given is the total amount of interest, \(\$178\text{,}\) which is also in dollars. The Visa had \(v\) dollars charged to it and accrues \(2\%\) interest. So \(0.02v\) is the dollar amount of interest that comes from using this card. Similarly, \(0.025m\) is the dollar amount of interest from using the Mastercard. Together:

\begin{equation*} 0.02(v\,\text{dollars})+0.025(m\,\text{dollars})=\$178 \end{equation*}

Or without units:

\begin{equation*} 0.02v+0.025m=178 \end{equation*}

As a system, we write:

\begin{equation*} \left\{ \begin{alignedat}{3} v\amp+{}\amp m\amp={}8400 \\ 0.02v\amp+{}\amp 0.025m\amp{}=178 \\ \end{alignedat} \right. \end{equation*}

To solve this system by subsitution, notice that it will be easier to solve for one of the variables in the first equation. We'll solve that equation for \(v\text{:}\)

\begin{align*} v+m\amp=8400\\ v+m\subtractright{m}\amp=8400\subtractright{m}\\ v\amp=8400-m \end{align*}

Now we will substitute \(8400-m\) for \(v\) in the second equation:

\begin{align*} 0.02v+0.025m\amp=178\\ 0.02(\substitute{8400-m})+0.025m\amp=178\\ 168-0.02m+0.025m\amp=178\\ 168+0.005m\amp=178\\ 0.005m\amp=10\\ \divideunder{0.005m}{0.005}\amp=\divideunder{10}{0.005}\\ m\amp=2000 \end{align*}

Lastly, we can determine the value of \(v\) by using the earlier equation where we isolated \(v\text{:}\)

\begin{align*} v\amp=8400-m\\ v\amp=8400-\substitute{2000}\\ v\amp=6400 \end{align*}

In summary, Greta charged \(\$6400\) to the Visa and \(\$2000\) to the Mastercard. We should check that these numbers work as solutions to our original system and that they make sense in context. (For instance, if one of these numbers were negative, or was something small like \(\$0.50\text{,}\) they wouldn't make sense as credit card debt.)

The next three examples are called mixture problems, because they involve mixing two quantities together to form a combination and we want to find out how much of each quantity to mix.

Example5.2.11Mixing Solutions with Two Different Concentrations

LaVonda is a meticulous bartender and she needs to serve \(600\) milliliters of Rob Roy, an alcoholic cocktail that is \(34\%\) alcohol by volume. The main ingredients are scotch that is \(42\%\) alcohol and vermouth that is \(18\%\) alcohol. How many milliliters of each ingredient should she mix together to make the concentration she needs?

Solution

The two unknowns are the quantities of each ingredient. Let \(s\) be the amount of scotch (in mL) and let \(v\) be the amount of vermouth (in mL).

One quantity given to us in the problem is 600 mL. Since this is the total volume of the mixed drink, we must have:

\begin{equation*} (s\,\text{mL})+(v\,\text{mL})=600\,\text{mL} \end{equation*}

Or without units:

\begin{equation*} s+v=600 \end{equation*}

To build the second equation, we have to think about the alcohol concentrations for the scotch, vermouth, and Rob Roy. It can be tricky to think about percentages like these correctly. One strategy is to focus on the amount (in mL) of alcohol being mixed. If we have \(s\) milliliters of scotch that is \(42\%\) alcohol, then \(0.42s\) is the actual amount (in mL) of alochol in that scotch. Similarly, \(0.18v\) is the amount of alcohol in the vermouth. And the final cocktail is 600 mL of liquid that is \(34\%\) alcohol, so it has \(0.34(600)=204\) milliliters of alcohol. All this means:

\begin{equation*} 0.42(s\,\text{mL})+0.18(v\,\text{mL})=204\,\text{mL} \end{equation*}

Or without units:

\begin{equation*} 0.42s+0.18v=204 \end{equation*}

So our system is:

\begin{equation*} \left\{ \begin{alignedat}{3} s\amp+{}\amp v\amp=600 \\ 0.42s\amp+{}\amp0.18v\amp=204 \end{alignedat} \right. \end{equation*}

To solve this system, we'll solve for \(s\) in the first equation:

\begin{align*} s+v\amp=600\\ s\amp=600-v \end{align*}

And then substitute \(s\) in the second equation with \(600-v\text{:}\)

\begin{align*} 0.42s+0.18v\amp=204\\ 0.42(\substitute{600-v})+0.18v\amp=204\\ 252-0.42v+0.18v\amp=204\\ 252-0.24v\amp=204\\ -0.24v\amp=-48\\ \divideunder{-0.24v}{-0.24}\amp=\divideunder{-48}{-0.24}\\ v\amp=200 \end{align*}

As a last step, we will determine \(s\) using the equation where we had isolated \(s\text{:}\)

\begin{align*} s\amp=600-v\\ s\amp=600-\substitute{200}\\ s\amp=400 \end{align*}

In summary, LaVonda needs to combine 400 mL of scotch with 200 mL of vermouth to create 600 mL of Rob Roy that is \(34\%\) alcohol by volume.

As a check for Example 5.2.11, we will use estimation to see that our solution is reasonable. Since LaVonda is making a \(34\%\) solution, she would need to use more of the \(42\%\) concentration than the \(18\%\) concentration, because \(34\%\) is closer to \(42\%\) than to \(18\%\text{.}\) This agrees with our answer because we found that she needed 400 mL of the \(42\%\) solution and 200 mL of the \(18\%\) solution. This is an added check that we have found reasonable answers.

Example5.2.12Mixing a Coffee Blend

A coffee shop manager wants to mix two different types of coffee beans to make a blend that sells for \(\$12.50\) per pound. They have some coffee beans from Columbia that sell for \(\$9.00\) per pound and some coffee beans from Honduras that sell for \(\$14.00\) per pound. How many pounds of each should they mix to make \(30\) pounds of the blend?

Solution

Before we begin, it may be helpful to try to estimate the solution. Let's compare the three prices. Since \(\$12.50\) is between the prices of \(\$9.00\) and \(\$14.00\text{,}\) this mixture is possible. Now we need to estimate the amount of each type needed. The price of the blend (\(\$12.50\) per pound) is closer to the higher priced beans (\(\$14.00\) per pound) than the lower priced beans (\(\$9.00\) per pound). So we will need to use more of that type. Keeping in mind that we need a total of \(30\) pounds, we roughly estimate \(20\) pounds of the \(\$14.00\) Honduran beans and \(10\) pounds of the \(\$9.00\) Columbian beans. How good is our estimate? Next we will solve this exercise exactly.

To set up our system of equations we define variables, letting \(C\) be the amount of Columbian coffee beans (in pounds) and \(H\) be the amount of Honduran coffee beans (in pounds).

The equations in our system will come from the total amount of beans and the total cost. The equation for the total amount of beans can be written as:

\begin{equation*} (C\,\text{lb})+(H\,\text{lb})=30\,\text{lb} \end{equation*}

Or without units:

\begin{equation*} C+H=30 \end{equation*}

To build the second equation, we have to think about the cost of all these beans. If we have \(C\) pounds of Columbian beans that cost \(\$9.00\) per pound, then \(9C\) is the cost of those beans in dollars. Similarly, \(14H\) is the cost of the Honduran beans. And the total cost is for \(30\) pounds of beans priced at \(\$12.50\) per pound, totaling \(12.5(30)=37.5\) dollars. All this means:

\begin{equation*} \left(9\,\tfrac{\text{dollars}}{\text{lb}}\right)(C\,\text{lb})+\left(14\,\tfrac{\text{dollars}}{\text{lb}}\right)(H\,\text{lb})=\left(12.50\,\tfrac{\text{dollars}}{\text{lb}}\right)(30\,\text{lb}) \end{equation*}

Or without units and carrying out the multiplication on the right:

\begin{equation*} 9C+14H=37.5 \end{equation*}

Now our system is:

\begin{equation*} \left\{ \begin{alignedat}{3} C\amp+{}\amp H\amp=30 \\ 9C\amp+{}\amp 14H\amp=12.50(30) \end{alignedat} \right. \end{equation*}

To solve the system, we'll solve the first equation for \(C\text{:}\)

\begin{align*} C+H\amp=30\\ C\amp=30-H \end{align*}

Next, we'll substitute \(C\) in the second equation with \(30-H\text{:}\)

\begin{align*} 9C+14H\amp=375\\ 9(\substitute{30-H})+14H\amp=375\\ 270-9H+14H\amp=375\\ 270+5H\amp=375\\ 5H\amp=105\\ H\amp=21 \end{align*}

Since \(H=21\text{,}\) we can conclude that \(C=9\text{.}\)

In summary, they need to mix \(21\) pounds of the Honduran coffee beans with \(9\) pounds of the Columbian coffee beans to create this blend. Our estimate at the beginning was pretty close, so we feel this answer is reasonable.

Subsection5.2.3Solving Special Systems of Equations with Substitution

Remember the two special cases we encountered when solving by graphing in Section 5.1? If the two lines represented by a system of equations have the same slope, then they might be separate lines that never meet, meaning the system has no solutions. Or they might coincide as the same line, in which case there are infinitely many solutions represented by all the points on that line. Let's see what happens when we use the substitution method on each of the special cases.

Example5.2.13A System with No Solution

Solve the system of equations using the substitution method:

\begin{align*} \left\{ \begin{aligned} y \amp= 2x-1 \\ 4x - 2y \amp= 3 \\ \end{aligned} \right. \end{align*}
Solution

Since the first equation is already solved for \(y\text{,}\) we will substitute \(2x-1\) for \(y\) in the second equation, and we have:

\begin{align*} 4x-2y\amp=3\\ 4x-2\substitute{(2x-1)}\amp=3\\ 4x-4x+2\amp=3\\ 2\amp=3 \end{align*}

Even though we were only intending to substitute away \(y\text{,}\) we ended up with an equation where there are no variables at all. This will happen whenever the lines have the same slope. This tells us the system represents either parallel or coinciding lines. Since \(2=3\) is false no matter what values \(x\) and \(y\) might be, there can be no solution to the system. So the lines are parallel and distinct. We write the solution set using the empty set symbol: the solution set is \(\emptyset\text{.}\)

To verify this, re-write the second equation, \(4x-2y=3\text{,}\) in slope-intercept form:

\begin{align*} 4x-2y\amp=3\\ 4x-2y\subtractright{4x}\amp=3\subtractright{4x}\\ -2y\amp=-4x+3\\ \divideunder{-2y}{-2}\amp=\divideunder{-4x+3}{-2}\\ y\amp=\frac{-4x}{-2}+\frac{3}{-2}\\ y\amp=2x-\frac{3}{2} \end{align*}

So the system is equivalent to:

\begin{align*} \left\{ \begin{aligned} y \amp = 2x-1 \\ y \amp = 2x-\frac{3}{2} \\ \end{aligned} \right. \end{align*}

Now it is easier to see that the two lines have the same slope but different \(y\)-intercepts. They are parallel and distinct lines, so the system has no solution.

Example5.2.14A System with Infinitely Many Solutions

Solve the system of equations using the substitution method:

\begin{align*} \left\{ \begin{aligned} y \amp=2x-1 \\ 4x-2y \amp=2 \\ \end{aligned} \right. \end{align*}
Solution

Since \(y=2x-1\text{,}\) we will substitute \(2x-1\) for \(y\) in the second equation and we have:

\begin{align*} 4x-2y\amp=2\\ 4x-2\substitute{(2x-1)}\amp=2\\ 4x-4x+2\amp=2\\ 2\amp=2 \end{align*}

Even though we were only intending to substitute away \(y\text{,}\) we ended up with an equation where there are no variables at all. This will happen whenever the lines have the same slope. This tells us the system represents either parallel or coinciding lines. Since \(2=2\) is true no matter what values \(x\) and \(y\) might be, the system equations are true no matter what \(x\) is, as long as \(y=2x-1\text{.}\)

So the lines coincide. We write the solution set as \(\{(x,y)\mid y=2x-1\}\text{.}\)

To verify this, re-write the second equation, \(4x-2y=2\text{,}\) in slope-intercept form:

\begin{align*} 4x-2y\amp=2\\ 4x-2y\subtractright{4x}\amp=2\subtractright{4x}\\ -2y\amp=-4x+2\\ \frac{-2y}{-2}\amp=\frac{-4x}{-2}+\frac{2}{-2}\\ y\amp=2x-1 \end{align*}

The system looks like:

\begin{align*} \left\{ \begin{alignedat}{4} y \amp {}={} \amp 2x-1 \\ y \amp {}={} \amp 2x-1 \\ \end{alignedat} \right. \end{align*}

Now it is easier to see that the two equations represent the same line. Every point on the line is a solution to the system, so the system has infinitely many solutions. The solution set is \(\{(x,y) \mid y=2x-1\}\text{.}\)

Subsection5.2.4Exercises

Solving System of Equations Using Substitution

1

Solve the following system of equations.

\(\left\{ \begin{aligned} {2y} \amp = {-2x}\\ {0} \amp = {-y} \end{aligned} \right.\)

2

Solve the following system of equations.

\(\left\{ \begin{aligned} {y} \amp = {-12}\\ {-3y} \amp = {-4x} \end{aligned} \right.\)

3

Solve the following system of equations.

\(\left\{ \begin{aligned} {-10} \amp = {4y+5x}\\ {0} \amp = {-y} \end{aligned} \right.\)

4

Solve the following system of equations.

\(\left\{ \begin{aligned} {0} \amp = {-4y - 16}\\ {-5y} \amp = {-4x} \end{aligned} \right.\)

5

Solve the following system of equations.

\(\left\{ \begin{aligned} {B} \amp = {-3a}\\ {B} \amp = {3a} \end{aligned} \right.\)

6

Solve the following system of equations.

\(\left\{ \begin{aligned} {r} \amp = {-6 - 5a}\\ {r} \amp = {18+a} \end{aligned} \right.\)

7

Solve the following system of equations.

\(\left\{ \begin{aligned} {c} \amp = {-12+5B}\\ {2c+2B} \amp = {-12} \end{aligned} \right.\)

8

Solve the following system of equations.

\(\left\{ \begin{aligned} {p} \amp = {11 - 3a}\\ {4a+5p} \amp = {22} \end{aligned} \right.\)

9

Solve the following system of equations.

\(\left\{ \begin{aligned} {a} \amp = {-13 - 3q}\\ {-4a+q} \amp = {-39} \end{aligned} \right.\)

10

Solve the following system of equations.

\(\left\{ \begin{aligned} {y} \amp = {-3x+28}\\ {-3y+5x} \amp = {-28} \end{aligned} \right.\)

11

Solve the following system of equations.

\(\left\{ \begin{aligned} {x} \amp = {18 - 5y}\\ {-x+4y} \amp = {-18} \end{aligned} \right.\)

12

Solve the following system of equations.

\(\left\{ \begin{aligned} {x} \amp = {11 - 2y}\\ {-3y+4x} \amp = {-11} \end{aligned} \right.\)

13

Solve the following system of equations.

\(\left\{ \begin{aligned} {17} \amp = {x+4y}\\ {5y - 3x} \amp = {0} \end{aligned} \right.\)

14

Solve the following system of equations.

\(\left\{ \begin{aligned} {r - 3b} \amp = {-28}\\ {42+b} \amp = {5r} \end{aligned} \right.\)

15

Solve the following system of equations.

\(\left\{ \begin{aligned} {C} \amp = {{\frac{5}{3}} - b}\\ {C} \amp = {4 - b} \end{aligned} \right.\)

16

Solve the following system of equations.

\(\left\{ \begin{aligned} {t} \amp = {-5+A}\\ {t} \amp = {A+2} \end{aligned} \right.\)

17

Solve the following system of equations.

\(\left\{ \begin{aligned} {5r - 2B+5} \amp = {0}\\ {10 - B} \amp = {-5r} \end{aligned} \right.\)

18

Solve the following system of equations.

\(\left\{ \begin{aligned} {-2} \amp = {2q - n}\\ {-3n+5q+2} \amp = {0} \end{aligned} \right.\)

19

Solve the following system of equations.

\(\left\{ \begin{aligned} {2x - 5y} \amp = {-2}\\ {4} \amp = {-3x} \end{aligned} \right.\)

20

Solve the following system of equations.

\(\left\{ \begin{aligned} {-4 - 4x} \amp = {0}\\ {5} \amp = {-2y - 3x} \end{aligned} \right.\)

21

Solve the following system of equations.

\(\left\{ \begin{aligned} {-2y+2} \amp = {5x}\\ {-5+y} \amp = {3x} \end{aligned} \right.\)

22

Solve the following system of equations.

\(\left\{ \begin{aligned} {3x} \amp = {4y+4}\\ {-4} \amp = {3y+x} \end{aligned} \right.\)

23

Solve the following system of equations.

\(\left\{ \begin{aligned} {-3q+2B} \amp = {1}\\ {-5B - 2} \amp = {-q} \end{aligned} \right.\)

24

Solve the following system of equations.

\(\left\{ \begin{aligned} {2m} \amp = {1 - A}\\ {3m+5A - 3} \amp = {0} \end{aligned} \right.\)

25

Solve the following system of equations.

\(\left\{ \begin{aligned} {b+{\frac{1}{3}}} \amp = {0}\\ {-2+{\frac{1}{5}}q} \amp = {-{\frac{2}{3}}b} \end{aligned} \right.\)

26

Solve the following system of equations.

\(\left\{ \begin{aligned} {0} \amp = {{\frac{3}{4}} - b - {\frac{3}{5}}x}\\ {0} \amp = {2 - x} \end{aligned} \right.\)

27

Solve the following system of equations.

\(\left\{ \begin{aligned} {{\frac{1}{2}}+{\frac{3}{2}}n} \amp = {0}\\ {{\frac{4}{5}}n} \amp = {C+3} \end{aligned} \right.\)

28

Solve the following system of equations.

\(\left\{ \begin{aligned} {-x} \amp = {-1+{\frac{3}{5}}y}\\ {{\frac{4}{5}}x+1} \amp = {0} \end{aligned} \right.\)

29

Solve the following system of equations.

\(\left\{ \begin{aligned} {0} \amp = {x+{\frac{2}{5}}+2y}\\ {{\frac{3}{5}}y} \amp = {-{\frac{4}{5}}x - 2} \end{aligned} \right.\)

30

Solve the following system of equations.

\(\left\{ \begin{aligned} {{\frac{3}{4}}y} \amp = {-4+x}\\ {{\frac{2}{5}}} \amp = {-y - x} \end{aligned} \right.\)

31

Solve the following system of equations.

\(\left\{ \begin{aligned} {-y - {\frac{5}{2}}x} \amp = {0}\\ {-{\frac{5}{4}}+y} \amp = {x} \end{aligned} \right.\)

32

Solve the following system of equations.

\(\left\{ \begin{aligned} {-{\frac{4}{5}}q} \amp = {1 - p}\\ {-2p - q} \amp = {{\frac{3}{2}}} \end{aligned} \right.\)

33

Solve the following system of equations.

\(\left\{ \begin{aligned} {-2y+4x} \amp = {-3}\\ {3x - 4y} \amp = {-5} \end{aligned} \right.\)

34

Solve the following system of equations.

\(\left\{ \begin{aligned} {x+3y} \amp = {-3}\\ {y - 3x} \amp = {-5} \end{aligned} \right.\)

35

Solve the following system of equations.

\(\left\{\begin{aligned} {-4x+3y} \amp = {{\frac{35}{44}}} \\ {-4x+5y} \amp = {{\frac{101}{44}}} \end{aligned}\right.\)

36

Solve the following system of equations.

\(\left\{\begin{aligned} {4x-3y} \amp = {-{\frac{99}{14}}} \\ {5x+3y} \amp = {{\frac{207}{14}}} \end{aligned}\right.\)

37

Solve the following system of equations.

\(\left\{\begin{aligned} {-{\frac{1}{5}}x-{\frac{1}{4}}y} \amp = {-{\frac{467}{800}}} \\ {{\frac{1}{3}}x+{\frac{1}{2}}y} \amp = {{\frac{87}{80}}} \end{aligned}\right.\)

38

Solve the following system of equations.

\(\left\{\begin{aligned} {{\frac{1}{4}}x+{\frac{1}{4}}y} \amp = {{\frac{103}{264}}} \\ {-{\frac{1}{4}}x+{\frac{1}{5}}y} \amp = {-{\frac{1}{66}}} \end{aligned}\right.\)

39

Solve the following system of equations.

\(\left\{\begin{aligned} {5x+y} \amp = {35} \\ {4x+2y} \amp = {28} \end{aligned}\right.\)

40

Solve the following system of equations.

\(\left\{\begin{aligned} {2x+5y} \amp = {-17} \\ {4x+4y} \amp = {8} \end{aligned}\right.\)

41

Solve the following system of equations.

\(\left\{\begin{aligned} {5x-4y} \amp = {-78} \\ {6x+2y} \amp = {-46} \end{aligned}\right.\)

42

Solve the following system of equations.

\(\left\{\begin{aligned} {-x+5y} \amp = {2} \\ {5x+2y} \amp = {-37} \end{aligned}\right.\)

43

Solve the following system of equations.

\(\left\{\begin{aligned} {-3x-5y} \amp = {55} \\ {-2x-4y} \amp = {42} \end{aligned}\right.\)

44

Solve the following system of equations.

\(\left\{\begin{aligned} {-x-2y} \amp = {-9} \\ {-2x-2y} \amp = {-6} \end{aligned}\right.\)

45

Solve the following system of equations.

\(\left\{\begin{aligned} {-2x} \amp = {0} \\ {2x-2y} \amp = {4} \end{aligned}\right.\)

46

Solve the following system of equations.

\(\left\{\begin{aligned} {2x-3y} \amp = {31} \\ {2x} \amp = {4} \end{aligned}\right.\)

47

Solve the following system of equations.

\(\left\{\begin{aligned} {4x+4y} \amp = 3 \\ {8x+8y} \amp = 3 \end{aligned}\right.\)

48

Solve the following system of equations.

\(\left\{\begin{aligned} {5x+2y} \amp = 2 \\ {15x+6y} \amp = 2 \end{aligned}\right.\)

49

Solve the following system of equations.

\(\left\{\begin{aligned} {5x+y} \amp = 2 \\ {-20x-4y} \amp = -8 \end{aligned}\right.\)

50

Solve the following system of equations.

\(\left\{\begin{aligned} {x+4y} \amp = 2 \\ {-3x-12y} \amp = -6 \end{aligned}\right.\)

Application Problems

51

A rectangle’s length is \(6\) feet longer than three times its width. The rectangle’s perimeter is \(132\) feet. Find the rectangle’s length and width.

The rectangle’s length is feet, and its width is feet.

52

A school fund raising event sold a total of \(209\) tickets and generated a total revenue of \({\$622.20}\text{.}\) There are two types of tickets: adult tickets and child tickets. Each adult ticket costs \({\$4.60}\text{,}\) and each child ticket costs \({\$1.95}\text{.}\)

Write and solve a system of equations to answer the following questions.

adult tickets and child tickets were sold.

53

Phone Company A charges a monthly fee of \({\$37.00}\text{,}\) and \({\$0.04}\) for each minute of talk time.

Phone Company B charges a monthly fee of \({\$25.00}\text{,}\) and \({\$0.09}\) for each minute of talk time.

Write and solve a system equation to answer the following questions.

These two companies would charge the same amount on a monthly bill when the talk time was minutes.

54

Company A’s revenue this fiscal year is \({\$802{,}000}\text{,}\) but its revenue is decreasing by \({\$9{,}000}\) each year.

Company B’s revenue this fiscal year is \({\$592{,}000}\text{,}\) and its revenue is increasing by \({\$5{,}000}\) each year.

Write and solve a system of equations to answer the following question.

After years, Company B will catch up with Company A in revenue.

55

A test has \(24\) problems, which are worth a total of \(150\) points. There are two types of problems in the test. Each multiple-choice problem is worth \(4\) points, and each short-answer problem is worth \(10\) points.

Write and solve a system equation to answer the following questions.

This test has multiple-choice problems and short-answer problems.

56

Priscilla invested a total of \({\$8{,}500}\) in two accounts. One account pays \(5\%\) interest annually; the other pays \(6\%\) interest annually. At the end of the year, Priscilla earned a total of \({\$480}\) in interest. How much money did Priscilla invest in each account?

Write and solve a system of equations to answer the following questions.

Priscilla invested in the \(5\%\) account.

Priscilla invested in the \(6\%\) account.

57

Jerry invested a total of \({\$14{,}000}\) in two accounts. After a year, one account lost \(6.3\%\text{,}\) while the other account gained \(2.5\%\text{.}\) In total, Jerry lost \({\$354}\text{.}\) How much money did Jerry invest in each account?

Write and solve a system of equations to answer the following questions.

Jerry invested in the account with \(6.3\%\) loss.

Jerry invested in the account with \(2.5\%\) gain.

58

Town A and Town B were located close to each other, and recently merged into one city. Town A had a population with \(12\%\) Hispanics. Town B had a population with \(8\%\) Hispanics. After the merge, the new city has a total of \(5000\) residents, with \(9.12\%\) Hispanics. How many residents did Town A and Town B used to have?

Write and solve a system equation to answer the following questions.

Town A used to have residents, and Town B used to have residents.

59

You poured some \(12\%\) alcohol solution and some \(6\%\) alcohol solution into a mixing container. Now you have \(480\) grams of \(8 \%\) alcohol solution. How many grams of \(12\%\) solution and how many grams of \(6 \%\) solution did you pour into the mixing container?

Write and solve a system equation to answer the following questions.

You mixed grams of \(12\%\) solution with grams of \(6\%\) solution.

60

The following table demonstrates the relation between interest rate, principal investment, and amount of interest. Fill in the missing entries with expressions or numbers.

Rate \(\times\) Principal \(=\) Interest
Solution 1 \(12\)% \(100\) \(12\)
Solution 2 \(64\)% \(500\)
Solution 3 \(70\)% \(41\)
Solution 4 \(7\)% \(280\)
Solution 5 \(4.7\)% \(260\)
Solution 6 \(5\)% \(x\)
Solution 7 \(2.6\)% \(5000-x\)
61

Dave invested a total of \({\$6{,}000}\) in two accounts. One account pays \(7\%\) interest annually; the other pays \(6\%\) interest annually. At the end of the year, Dave earned a total of \({\$380}\) in interest. How much money did Dave invest in each account?

Dave invested in the \(7\%\) account.

Dave invested in the \(6\%\) account.

62

Lily invested a total of \({\$61{,}000}\) in two accounts. One account pays \(5.4\%\) interest annually; the other pays \(6.4\%\) interest annually. At the end of the year, Lily earned a total interest of \({\$3{,}494}\text{.}\) How much money did Lily invest in each account?

Lily invested in the \(5.4\%\) account.

Lily invested in the \(6.4\%\) account.

63

Jessica invested a total of \({\$7{,}500}\) in two accounts. One account pays \(3\%\) interest annually; the other pays \(6\%\) interest annually. At the end of the year, Jessica earned the same amount of interest from both accounts. How much money did Jessica invest in each account?

Jessica invested in the \(3\%\) account.

Jessica invested in the \(6\%\) account.

64

Dave invested a total of \({\$61{,}000}\) in two accounts. One account pays \({7.8}\%\) interest annually; the other pays \({2.2}\%\) interest annually. At the end of the year, Dave earned the same amount of interest from both accounts. How much money did Dave invest in each account?

Dave invested in the \({7.8}\%\) account.

Dave invested in the \({2.2}\%\) account.

65

Randi invested a total of \({\$14{,}000}\) in two accounts. After a year, one account had earned \(12.6\%\text{,}\) while the other account had lost \(7.4\%\text{.}\) In total, Randi had a net gain of \({\$564}\text{.}\) How much money did Randi invest in each account?

Randi invested in the account that grew by \(12.6\%\text{.}\)

Randi invested in the account that fell by \(7.4\%\text{.}\)

66

The following table demonstrates the relation between the concentration (by mass) of alcohol in a solution, the mass of the solution, and the mass of the pure alcohol in that solution. Fill in the missing entries with expressions or numbers.

Percent of Weight of Weight of Pure
Alcohol \(\times\) Solution (in grams) \(=\) Alcohol (in grams)
Solution 1 \(92\%\) \(100\) \(92\)
Solution 2 \(27\%\) \(400\)
Solution 3 \(70\%\) \(26\)
Solution 4 \(2\%\) \(230\)
Solution 5 \(2.5\%\) \(220\)
Solution 6 \(8\%\) \(x\)
Solution 7 \(5.2\%\) \(300-x\)
67

You’ve poured \(200\) grams of a \(2\%\) (by mass) alcohol solution into a large glass container. You have plenty of \(6\%\) alcohol solution. You need to make some \(4.4\%\) alcohol solution. How many grams of \(6\%\) solution do you have to add to the glass container to end up with a \(4.4\%\) alcohol solution?

You need to add grams of \(6\%\) solution to end up with a \(4.4\%\) alcohol solution.

68

You’ve poured some \(8\%\) (by mass) alcohol solution and some \(6\%\) alcohol solution into a large glass mixing container. Now you have \(600\) grams of \(6.4\%\) alcohol solution. How many grams of \(8\%\) solution and how many grams of \(6\%\) solution did you pour into the mixing container?

You poured grams of \(8\%\) solution and grams of \(6\%\) solution into the mixing container.

69

You’ve poured \(1200\) grams of a \(15\%\) (by mass) alcohol solution into a large glass container. You have plenty of \(100\%\) pure alcohol on hand. You need to make some \(20\%\) alcohol solution. How many grams of pure alcohol do you have to add to the glass container to end up with a \(20\%\) alcohol solution?

You need to add grams of pure alcohol to end up with a \(20\%\) alcohol solution.

70

Town A and Town B are located closed to each other, and recently merged into one city. Town A used to have \(1400\) residents, with \(6\%\) Hispanics. Town B used to have \(4\%\) Hispanics. After the merge, the new city has \(4.4\%\) Hispanics. How many residents does the new city have now?

The new city has residents.

71

A coffee shop has \(35\) pounds of dark coffee, which sells for \({\$14.70}\) per pound. It also has some light coffee, which sells for \({\$13.50}\) per pound. The coffee shop plans to mix some light coffee into the dark coffee, and sell the mixture for \({\$13.85}\) per pound. How many pounds of light coffee should be mixed in?

To make coffee worth \({\$13.85}\text{,}\) the coffee shop needs to mix pounds of light coffee with the dark coffee.

72

A store has some beans selling for \({\$2.90}\) per pound, and some vegetables selling for \({\$3.50}\) per pound. The store plans to use them to produce \(18\) pounds of mixture and sell for \({\$3.17}\) per pound. How many pounds of beans and how many pounds of vegetables should be used?

To produce \(18\) pounds of mixture, the store should use pounds of beans and pounds of vegetables.

73

Town A and Town B were located close to each other, and recently merged into one city. Town A had a population with \(6\%\) Hispanics. Town B had a population with \(12\%\) Hispanics. After the merge, the new city has a total of \(3600\) residents, with \(10\%\) Hispanics. How many residents did Town A and Town B used to have?

Town A used to have residents, and Town B used to have residents.

74

Town A and Town B are located closed to each other, and recently merged into one city. Town A used to have \(3000\) residents, with \(4\%\) African Americans. Town B used to have \(6\%\) African Americans. After the merge, the new city has \(5.25\%\) African Americans. How many residents does the new city have now?

The new city has residents.