## Section 4.1 Solving Systems of Linear Equations by Graphing

Ā¶###### Objectives: PCC Course Content and Outcome Guide

We have learned how to graph a line given its equation. In this section, we will learn what a *system* of *two* linear equations is, and how to use graphing to solve such a system.

### Subsection 4.1.1 Solving Systems of Equations by Graphing

###### Example 4.1.2.

Fabiana and David are running at constant speeds in parallel lanes on a track. David starts out ahead of Fabiana, but Fabiana is running faster. We want to determine when Fabiana will catch up with David. Let's start by looking at the graph of each runner's distance over time, in FigureĀ 4.1.3.

Each of the two lines has an equation, as discussed in ChapterĀ 3. The line representing David appears to have \(y\)-intercept \((0,4)\) and slope \(\frac{4}{3}\text{,}\) so its equation is \(y=\frac{4}{3}t+4\text{.}\) The line representing Fabiana appears to have \(y\)-intercept \((0,0)\) and slope \(2\text{,}\) so its equation is \(y=2t\text{.}\)

When these two equations are together as a package, we have what is called a āsystem of linear equationsā:

The large left brace indicates that this is a collection of two distinct equations, not one equation that was somehow algebraically manipulated into an equivalent equation.

As we can see in FigureĀ 4.1.3, the graphs of the two equations cross at the point \((6,12)\text{.}\) We refer to the point \((6,12)\) as the solution to this system of linear equations. To denote the solution set, we write \(\{(6,12)\}\text{.}\) It's more valuable to interpret these numbers in context whenever possible: it took \(6\) seconds for the two runners to meet up, and when they met they were \(12\) meters up the track.

###### Definition 4.1.4. System of Linear Equations.

A system of linear equations is any pairing of two (or more) linear equations. A solution to a system of linear equations is any point that is a solution for all of the equations in the system. The solution set to a system of linear equations is the collection of all solutions to the system.

###### Remark 4.1.5.

In ExampleĀ 4.1.2, we stated that the solution was the point \((6,12)\text{.}\) It makes sense to write this as an ordered pair when we're given a graph. In some cases when we have no graph, particularly when our variables are not \(x\) and \(y\text{,}\) it might not be clear which variable ācomes firstā and we won't be able to write an ordered pair. Nevertheless, given the context we can write meaningful summary statements.

###### Example 4.1.6.

Determine the solution to the system of equations graphed in FigureĀ 4.1.7.

The two lines intersect where \(x=-3\) and \(y=-1\text{,}\) so the solution is the point \((-3,-1)\text{.}\) We write the solution set as \(\{(-3,-1)\}\text{.}\)

###### Checkpoint 4.1.8.

Now let's look at an example where *we* need to make a graph to find the solution.

###### Example 4.1.9.

Solve the following system of equations by graphing:

Notice that each of these equations is written in slope-intercept form. The first equation, \(y=\frac{1}{2}x+4\text{,}\) is a linear equation with a slope of \(\frac{1}{2}\) and a \(y\)-intercept of \((0,4)\text{.}\) The second equation, \(y=-x-5\text{,}\) is a linear equation with a slope of \(-1\) and a \(y\)-intercept of \((0,-5)\text{.}\) We'll use this information to graph both lines.

It appears that the two lines intersect where \(x=-6\) and \(y=1\text{,}\) so the solution of the system of equations would be the point \((-6,1)\text{.}\) We should check this with the two original equations.

This verifies that \((-6,1)\) is the solution, and we write the solution set as \(\{(-6,1)\}\text{.}\)

###### Example 4.1.11.

Solve the following system of equations by graphing:

Since both line equations are given in standard form, we'll graph each one by finding the intercepts. Recall that to find the \(x\)-intercept of each equation, replace \(y\) with \(0\) and solve for \(x\text{.}\) Similarly, to find the \(y\)-intercept of each equation, replace \(x\) with \(0\) and solve for \(y\text{.}\)

For our first linear equation, we have:

So the intercepts are \((-12,0)\) and \((0,4)\text{.}\)

For our second linear equation, we have:

So the intercepts are \(\left(\frac{3}{2},0\right)\) and \((0,1)\text{.}\)

Now we can graph each line by plotting the intercepts and connecting these points:

It appears that the solution of the system of equations is the point of intersection of those two lines, which is \((-3,3)\text{.}\) It's important to check this is correct, because when making a hand-drawn graph, it would be easy to be off by a little bit. To check, we can substitute the values of \(x\) and \(y\) from the point \((-3,3)\) into each equation:

So we have checked that \((-3,3)\) is indeed the solution for the system. We write the solution set as \(\{(-3,3)\}\text{.}\)

###### Example 4.1.13.

A college has a north campus and a south campus. The north campus has \(18{,}000\) students, and the south campus has \(4{,}000\) students. In the past five years, the north campus lost \(4{,}000\) students, and the south campus gained \(3{,}000\) students. If these trends continue, in how many years would the two campuses have the same number of students? Write and solve a system of equations modeling this problem.

Since all the given student counts are in the thousands, we make the decision to measure student population in thousands. So for instance, the north campus starts with a student population of \(18\) (thousand students).

The north campus lost \(4\) thousand students in \(5\) years. So it is losing students at a rate of \(\frac{4\text{ thousand}}{5\text{ year}}\text{,}\) or \(\frac{4}{5}\,\frac{\text{thousand}}{\text{year}}\text{.}\) This rate of change should be interpreted as a negative number, because the north campus is losing students over time. So we have a linear model with starting value \(18\) thousand students, and a slope of \(-\frac{4}{5}\) thousand students per year. In other words,

where \(y\) stands for the number of students in thousands, and \(t\) stands for the number of years into the future.

Similarly, the number of students at the south campus can be modeled by \(y=\frac{3}{5}t+4\text{.}\) Now we have a system of equations:

We will graph both lines using their slopes and \(y\)-intercepts.

According to the graph, the lines intersect at \((10,10)\text{.}\) So if the trends continue, both campuses will have \(10{,}000\) students \(10\) years from now.

###### Example 4.1.15.

Solve the following system of equations by graphing:

Since both line equations are given in point-slope form, we can start by graphing the point indicated in each equation and use the slope to determine the rest of the line.

For our first equation, \(y=3(x-2)+1\text{,}\) the point indicated in the equation is \((2,1)\) and the slope is \(3\text{.}\)

For our second equation, \(y=-\frac{1}{2}(x+1)-1\text{,}\) the point indicated in the equation is \((-1,-1)\) and the slope is \(-\frac{1}{2}\text{.}\)

Now we can graph each line by plotting the points and using their slopes.

It appears that the solution of the system of equations is the point of intersection of those two lines, which is \((1,-2)\text{.}\) It's important to check this is correct, because when making a hand-drawn graph, it would be easy to be off by a little bit. To check, we can substitute the values of \(x\) and \(y\) from the point \((1,-2)\) into each equation:

So we have checked that \((2,-1)\) is indeed the solution for the system. We write the solution set as \(\{(2,-1)\}\text{.}\)

### Subsection 4.1.2 Special Systems of Equations

Ā¶Recall that when we solved linear equations in one variable, there were two special cases discussed in detail in SectionĀ 2.4. In one special case, like with the equation \(x=x+1\text{,}\) there is no solution. And in the another case, like with the equation \(x=x\text{,}\) there are infinitely many solutions. When solving systems of equations in two variables, we have similar special cases to consider.

###### Example 4.1.17. Parallel Lines.

Let's look at the graphs of two lines with the same slope, \(y=2x-4\) and \(y=2x+1\text{:}\)

For this system of equations, what is the solution? Since the two lines have the same slope they are parallel lines and will never intersect. This means that there is *no solution* to this system of equations. We write the solution set as \(\emptyset\text{.}\)

The symbol \(\emptyset\) is a special symbol that represents the empty set, a *set* that has no numbers in it which can also be written simply as \(\{\text{ }\}\text{.}\) This symbol is *not* the same thing as the number zero. The *number* of eggs in an empty egg carton is zero whereas the empty carton itself could represent the empty set. The symbols for the empty set and the number zero may look similar depending on how you write the number zero, so try to keep the concepts separate.

###### Example 4.1.19. Coinciding Lines.

Next we'll look at the other special case. Let's start with this system of equations:

To solve this system of equations, we want to graph each line. The first equation is in slope-intercept form and can be graphed easily using its slope of \(2\) and its \(y\)-intercept of \((0,-4)\text{.}\)

The second equation, \(6x-3y=12\text{,}\) can either be graphed by solving for \(y\) and using the slope-intercept form or by finding the intercepts. If we use the intercept method, we'll find that this line has an \(x\)-intercept of \((2,0)\) and a \(y\)-intercept of \((0,-4)\text{.}\) When we graph both lines we get FigureĀ 4.1.20.

Now we can see these are actually the *same* line, or coinciding lines. To determine the solution to this system, we'll note that they overlap everywhere. This means that we have an infinite number of solutions: *all* points that fall on the line. It may be enough to report that there are infinitely many solutions. In order to be more specific, all we can do is say that any ordered pair \((x,y)\) satisfying the line equation is a solution. In set-builder notation, we would write \(\{(x,y)\mid y=2x-4\}\text{.}\)

###### Remark 4.1.21.

In ExampleĀ 4.1.19, what would have happened if we had decided to convert the second line equation into slope-intercept form?

This is the literally the same as the first equation in our system. This is a different way to show that these two equations are equivalent and represent the same line. Any time we try to solve a system where the equations are equivalent, we'll have an infinite number of solutions.

###### Warning 4.1.22.

Notice that for a system of equations with infinite solutions like ExampleĀ 4.1.19, we didn't say that *every* point was a solution. Rather, *every point that falls on that line* is a solution. It would be incorrect to state this solution set as āall real numbersā or as āall ordered pairs.ā

### Reading Questions 4.1.3 Reading Questions

###### 1.

What is the purpose of the one big left brace in a system of two equations?

###### 2.

When you find a solution to a system of two linear equations in two variables, why should you check the solution? Would it be good enough to only substitute the numbers into *one* of the original two equations?

###### 3.

Suppose you have a system of two linear equations, and you know the system has exactly one solution. What can you say about the slopes of the two lines that the two equations define?

### Exercises 4.1.4 Exercises

###### Warmup and Review

###### 1.

Find the lineās slope and \(y\)-intercept.

A line has equation \(y={3}x+5\text{.}\)

This lineās slope is .

This lineās \(y\)-intercept is .

###### 2.

Find the lineās slope and \(y\)-intercept.

A line has equation \(y={4}x+1\text{.}\)

This lineās slope is .

This lineās \(y\)-intercept is .

###### 3.

Find the lineās slope and \(y\)-intercept.

A line has equation \(y=-x - 1\text{.}\)

This lineās slope is .

This lineās \(y\)-intercept is .

###### 4.

Find the lineās slope and \(y\)-intercept.

A line has equation \(y=-x+1\text{.}\)

This lineās slope is .

This lineās \(y\)-intercept is .

###### 5.

Find the lineās slope and \(y\)-intercept.

A line has equation \(\displaystyle{ y= -\frac{6x}{5} +7 }\text{.}\)

This lineās slope is .

This lineās \(y\)-intercept is .

###### 6.

Find the lineās slope and \(y\)-intercept.

A line has equation \(\displaystyle{ y= -\frac{8x}{7} +4 }\text{.}\)

This lineās slope is .

This lineās \(y\)-intercept is .

###### 7.

Find the lineās slope and \(y\)-intercept.

A line has equation \(\displaystyle{ y= \frac{x}{10} - 4 }\text{.}\)

This lineās slope is .

This lineās \(y\)-intercept is .

###### 8.

Find the lineās slope and \(y\)-intercept.

A line has equation \(\displaystyle{ y= \frac{x}{10} +10 }\text{.}\)

This lineās slope is .

This lineās \(y\)-intercept is .

###### 9.

Graph the equation \(y=-3x\text{.}\)

###### 10.

Graph the equation \(y=\frac{1}{4}x\text{.}\)

###### 11.

Graph the equation \(y=\frac{2}{3}x+4\text{.}\)

###### 12.

Graph the equation \(y=-2x+5\text{.}\)

###### 13.

Solve the linear equation for \(y\text{.}\)

\({16x+2y}={-8}\)

###### 14.

Solve the linear equation for \(y\text{.}\)

\({6x-2y}={28}\)

###### 15.

Solve the linear equation for \(y\text{.}\)

\({8x+4y}={40}\)

###### 16.

Solve the linear equation for \(y\text{.}\)

\({3x+5y}={45}\)

###### Checking Solutions for System of Equations

###### 17.

Decide whether \((1,-3)\) is a solution to the system of equations:

The point \((1,-3)\)

is

is not

###### 18.

Decide whether \((2,4)\) is a solution to the system of equations:

The point \((2,4)\)

is

is not

###### 19.

Decide whether \((3,1)\) is a solution to the system of equations:

The point \((3,1)\)

is

is not

###### 20.

Decide whether \((4,-3)\) is a solution to the system of equations:

The point \((4,-3)\)

is

is not

###### 21.

Decide whether \(\left({{\frac{8}{5}}},{{\frac{4}{5}}}\right)\) is a solution to the system of equations:

The point \(\left({{\frac{8}{5}}},{{\frac{4}{5}}}\right)\)

is

is not

###### 22.

Decide whether \(\left({{\frac{7}{2}}},{{\frac{9}{2}}}\right)\) is a solution to the system of equations:

The point \(\left({{\frac{7}{2}}},{{\frac{9}{2}}}\right)\)

is

is not

###### Using a Graph to Solve a System

Use a graph to solve the system of equations.

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###### Determining the Number of Solutions in a System of Equations

###### 43.

Simply by looking at this system of equations, decide the number of solutions it has.

The system has

no solution

one solution

infinitely many solutions

###### 44.

Simply by looking at this system of equations, decide the number of solutions it has.

The system has

no solution

one solution

infinitely many solutions

###### 45.

Without graphing this system of equations, decide the number of solutions it has.

The system has

no solution

one solution

infinitely many solutions

###### 46.

Without graphing this system of equations, decide the number of solutions it has.

The system has

no solution

one solution

infinitely many solutions

###### 47.

Without graphing this system of equations, decide the number of solutions it has.

The system has

no solution

one solution

infinitely many solutions

###### 48.

Without graphing this system of equations, decide the number of solutions it has.

The system has

no solution

one solution

infinitely many solutions

###### 49.

Simply by looking at this system of equations, decide the number of solutions it has.

The system has

no solution

one solution

infinitely many solutions

###### 50.

Simply by looking at this system of equations, decide the number of solutions it has.

The system has

no solution

one solution

infinitely many solutions