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Section 13.6 Additional Practice Related to Systems of Linear Equations

Exercises Exercises

Determine the solution to each system of equations after first graphing each line in the system.

1.

\(\left\{ \begin{aligned} y\amp=\frac{3}{2}x-7\\ y\amp=-2x+7\\ \end{aligned} \right.\)

Solution

The solution is \((4,-1)\text{.}\)

2.

\(\left\{ \begin{aligned} 2x-y\amp=-11\\ -4x+3y\amp=21\\ \end{aligned} \right.\)

Solution

The solution is \((-6,-1)\text{.}\)

3.

\(\left\{ \begin{aligned} y\amp=-\frac{1}{3}x+4\\ 2x+6y\amp=12\\ \end{aligned} \right.\)

Solution

There are no solutions to the given system of equations

Use the method of substitution so solve each of the following system of equations.

4.

\(\left\{ \begin{aligned} x\amp=\frac{2}{5}y-14\\ 3x+5y\amp=20\\ \end{aligned} \right.\)

Solution

The solution is \((-10,10)\text{.}\)

5.

\(\left\{ \begin{aligned} y\amp=-\frac{7}{3}x+\frac{5}{6}\\ 14x+6y\amp=5\\ \end{aligned} \right.\)

Solution

The two equations graph to the same line. Every point on that line is a solution to the system of equations.

6.

\(\left\{ \begin{aligned} 3x-y\amp=23\\ -2x-11y\amp=8\\ \end{aligned} \right.\)

Solution

The solution is \((7,-2)\text{.}\)

7.

\(\left\{ \begin{aligned} 5x+7y\amp=-27\\ -2x+3y\amp=-24\\ \end{aligned} \right.\)

Solution

The solution is \((3,-6)\text{.}\)

8.

\(\left\{ \begin{aligned} \frac{1}{3}x-\frac{2}{7}y\amp=12\\ \frac{3}{2}x+\frac{3}{5}y\amp=-12\\ \end{aligned} \right.\)

Solution

The solution is \((6,-35)\text{.}\)

9.

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

Solution

The solution is \(\left(\frac{20}{13},\frac{41}{13}\right)\text{.}\)

Use the elimination method so solve each of the following system of equations.

10.

\(\left\{ \begin{aligned} 2x-5y\amp=49\\ -2x-5y\amp=1\\ \end{aligned} \right.\)

Solution

The solution is \((12,-5)\text{.}\)

11.

\(\left\{ \begin{aligned} -4x+\frac{3}{2}y\amp=-22\\ 3x+5y\amp=41\\ \end{aligned} \right.\)

Solution

The solution is \((7,4)\text{.}\)

12.

\(\left\{ \begin{aligned} y\amp=\frac{4}{3}x+4\\ 4x-3y\amp=21\\ \end{aligned} \right.\)

Solution

There are no solutions to the given system of equations.

Application Problems

13.

At the start of the year Nguyen had $15,000 invested in two separate IRA accounts - one a traditional IRA, the other a Roth IRA. Over the course of the year, the traditional IRA balance grew by 21% and the the Roth IRA balance grew by 17%. Nguyen made no deposits to nor withdraws from either account over the course of the year. At the end of the year the total balance of the two accounts was $17,718. How much did Nguyen have invested in each account at the beginning of the year?

Solution

At the beginning of the year Nguyen had $4,200 invested in the traditional IRA and $10,800 invested in the Roth IRA.

14.

Jamal is tasked with creating an acidic solution in his Chemistry lab. He needs to make 3 liters of a solution that is 35% Sulfuric acid and 65% water. He has two previously made solutions to work with. One is 20% acid and 80% water, the other is 40% acid and 60% water. How much of each of the preexisting solutions should Jamal use to create his new solution?

Solution

Jamal needs to mix \(.75\) liters of the 20% acid solution with 2.25 liters of the 40% acid solution.

15.

While on route, there are two speeds associated with a jet. One is the airspeed - the speed the jet would fly relative to the ground if there were no wind. The other is is the ground speed - the speed the jet actually moves relative to the ground. The wind due to the jet stream increases the ground speed of eastbound flights and decreases the ground speed of west bound flights, For example, if the airspeed is 400 mph and the wind speed is 50 mph, then the actual ground speed of an eastbound flight is 450 mph and the ground speed of a west bound flight is 350 mph. (This is a simplification of the specific impact of the jet stream, but reflective of the general concept.)

One afternoon two 747s are flying between San Francisco and Denver, one headed west (to San Francisco), the other headed east (to Denver). For 2.5 hours the two planes maintained the same constant airspeed. Over that period the eastbound flight traveled 1665 miles while the westbound flight only traveled 1185 miles. Determine the airspeed and wind speed for those 2.5 hours.

Solution

The airspeed is 570 mph and the wind speed is 96 mph.

16.

Olivia and her wife Jackie like to go running at the local high school track. On average, Jackie maintains a pace that is 30% greater than the pace maintained by Olivia. Over a thirty minute period, Jackie runs .825 miles farther than Olivia. Determine the speed (mi/hr) of each runner, assuming that they each maintain a constant speed for the entire thirty minute period.

Solution

Jackie's constant speed is 7.15 mi/hr and Olivia's constant speed is 5.5 mi/hr.