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Mathematics LibreTexts

5.2E: Exercises

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    30409
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    Practice Makes Perfect

    Solve a System of Equations by Substitution

    In the following exercises, solve the systems of equations by substitution.

    Exercise \(\PageIndex{1}\)

    \(\left\{\begin{array}{l}{2 x+y=-4} \\ {3 x-2 y=-6}\end{array}\right.\)

    Answer

    \((−2,0)\)

    Exercise \(\PageIndex{2}\)

    \(\left\{\begin{array}{l}{2 x+y=-2} \\ {3 x-y=7}\end{array}\right.\)

    Exercise \(\PageIndex{3}\)

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

    Answer

    \((7,6)\)

    Exercise \(\PageIndex{4}\)

    \(\left\{\begin{array}{l}{x-3 y=-9} \\ {2 x+5 y=4}\end{array}\right.\)

    Exercise \(\PageIndex{5}\)

    \(\left\{\begin{array}{l}{5 x-2 y=-6} \\ {y=3 x+3}\end{array}\right.\)

    Answer

    \((0,3)\)

    Exercise \(\PageIndex{6}\)

    \(\left\{\begin{array}{l}{-2 x+2 y=6} \\ {y=-3 x+1}\end{array}\right.\)

    Exercise \(\PageIndex{7}\)

    \(\left\{\begin{array}{l}{2 x+3 y=3} \\ {y=-x+3}\end{array}\right.\)

    Answer

    \((6,−3)\)

    Exercise \(\PageIndex{8}\)

    \(\left\{\begin{array}{l}{2 x+5 y=-14} \\ {y=-2 x+2}\end{array}\right.\)

    Exercise \(\PageIndex{9}\)

    \(\left\{\begin{array}{l}{2 x+5 y=1} \\ {y=\frac{1}{3} x-2}\end{array}\right.\)

    Answer

    \((3,−1)\)

    Exercise \(\PageIndex{10}\)

    \(\left\{\begin{array}{l}{3 x+4 y=1} \\ {y=-\frac{2}{5} x+2}\end{array}\right.\)

    Exercise \(\PageIndex{11}\)

    \(\left\{\begin{array}{l}{3 x-2 y=6} \\ {y=\frac{2}{3} x+2}\end{array}\right.\)

    Answer

    \((6,6)\)

    Exercise \(\PageIndex{12}\)

    \(\left\{\begin{array}{l}{-3 x-5 y=3} \\ {y=\frac{1}{2} x-5}\end{array}\right.\)

    Exercise \(\PageIndex{13}\)

    \(\left\{\begin{array}{l}{2 x+y=10} \\ {-x+y=-5}\end{array}\right.\)

    Answer

    \((5,0)\)

    Exercise \(\PageIndex{14}\)

    \(\left\{\begin{array}{l}{-2 x+y=10} \\ {-x+2 y=16}\end{array}\right.\)

    Exercise \(\PageIndex{15}\)

    \(\left\{\begin{array}{l}{3 x+y=1} \\ {-4 x+y=15}\end{array}\right.\)

    Answer

    \((−2,7)\)

    Exercise \(\PageIndex{16}\)

    \(\left\{\begin{array}{l}{x+y=0} \\ {2 x+3 y=-4}\end{array}\right.\)

    Exercise \(\PageIndex{17}\)

    \(\left\{\begin{array}{l}{x+3 y=1} \\ {3 x+5 y=-5}\end{array}\right.\)

    Answer

    \((−5,2)\)

    Exercise \(\PageIndex{18}\)

    \(\left\{\begin{array}{l}{x+2 y=-1} \\ {2 x+3 y=1}\end{array}\right.\)

    Exercise \(\PageIndex{19}\)

    \(\left\{\begin{array}{l}{2 x+y=5} \\ {x-2 y=-15}\end{array}\right.\)

    Answer

    \((−1,7)\)

    Exercise \(\PageIndex{20}\)

    \(\left\{\begin{array}{l}{4 x+y=10} \\ {x-2 y=-20}\end{array}\right.\)

    Exercise \(\PageIndex{21}\)

    \(\left\{\begin{array}{l}{y=-2 x-1} \\ {y=-\frac{1}{3} x+4}\end{array}\right.\)

    Answer

    \((−3,5)\)

    Exercise \(\PageIndex{22}\)

    \(\left\{\begin{array}{l}{y=x-6} \\ {y=-\frac{3}{2} x+4}\end{array}\right.\)

    Exercise \(\PageIndex{23}\)

    \(\left\{\begin{array}{l}{y=2 x-8} \\ {y=\frac{3}{5} x+6}\end{array}\right.\)

    Answer

    \((10, 12)\)

    Exercise \(\PageIndex{24}\)

    \(\left\{\begin{array}{l}{y=-x-1} \\ {y=x+7}\end{array}\right.\)

    Exercise \(\PageIndex{25}\)

    \(\left\{\begin{array}{l}{4 x+2 y=8} \\ {8 x-y=1}\end{array}\right.\)

    Answer

    \(\left(\frac{1}{2}, 3\right)\)

    Exercise \(\PageIndex{26}\)

    \(\left\{\begin{array}{l}{-x-12 y=-1} \\ {2 x-8 y=-6}\end{array}\right.\)

    Exercise \(\PageIndex{27}\)

    \(\left\{\begin{array}{l}{15 x+2 y=6} \\ {-5 x+2 y=-4}\end{array}\right.\)

    Answer

    \(\left(\frac{1}{2},-\frac{3}{4}\right)\)

    Exercise \(\PageIndex{28}\)

    \(\left\{\begin{array}{l}{2 x-15 y=7} \\ {12 x+2 y=-4}\end{array}\right.\)

    Exercise \(\PageIndex{29}\)

    \(\left\{\begin{array}{l}{y=3 x} \\ {6 x-2 y=0}\end{array}\right.\)

    Answer

    Infinitely many solutions. The two equations represent the same line.
    The solution set is: \(\big\{ (x,y)\, | \,y = 3 x \big\}\)
    All points that are solutions of the equation \(y=3x\) are solutions of this system.

    Exercise \(\PageIndex{30}\)

    \(\left\{\begin{array}{l}{x=2 y} \\ {4 x-8 y=0}\end{array}\right.\)

    Exercise \(\PageIndex{31}\)

    \(\left\{\begin{array}{l}{2 x+16 y=8} \\ {-x-8 y=-4}\end{array}\right.\)

    Answer

    Infinitely many solutions. The two equations represent the same line.
    The solution set is: \(\big\{ (x,y) \,| \,2 x +16 y = 8 \big\}\)
    All points that are solutions of the equation \(2 x +16 y = 8 \) are solutions of this system.

    Exercise \(\PageIndex{32}\)

    \(\left\{\begin{array}{l}{15 x+4 y=6} \\ {-30 x-8 y=-12}\end{array}\right.\)

    Exercise \(\PageIndex{33}\)

    \(\left\{\begin{array}{l}{y=-4 x} \\ {4 x+y=1}\end{array}\right.\)

    Answer

    No solution

    Exercise \(\PageIndex{34}\)

    \(\left\{\begin{array}{l}{y=-\frac{1}{4} x} \\ {x+4 y=8}\end{array}\right.\)

    Exercise \(\PageIndex{35}\)

    \(\left\{\begin{array}{l}{y=\frac{7}{8} x+4} \\ {-7 x+8 y=6}\end{array}\right.\)

    Answer

    No solution

    Exercise \(\PageIndex{36}\)

    \(\left\{\begin{array}{l}{y=-\frac{2}{3} x+5} \\ {2 x+3 y=11}\end{array}\right.\)

    Solve Applications of Systems of Equations by Substitution

    In the following exercises, translate to a system of equations and solve.

    Exercise \(\PageIndex{37}\)

    The sum of two numbers is 15. One number is 3 less than the other. Find the numbers.

    Answer

    The numbers are 6 and 9.

    Exercise \(\PageIndex{38}\)

    The sum of two numbers is 30. One number is 4 less than the other. Find the numbers.

    Exercise \(\PageIndex{39}\)

    The sum of two numbers is −26. One number is 12 less than the other. Find the numbers.

    Answer

    The numbers are −7 and −19.

    Exercise \(\PageIndex{40}\)

    The perimeter of a rectangle is 50. The length is 5 more than the width. Find the length and width.

    Exercise \(\PageIndex{41}\)

    The perimeter of a rectangle is 60. The length is 10 more than the width. Find the length and width.

    Answer

    The length is 20 and the width is 10.

    Exercise \(\PageIndex{42}\)

    The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width.

    Exercise \(\PageIndex{43}\)

    The perimeter of a rectangle is 84. The length is 10 more than three times the width. Find the length and width.

    Answer

    The length is 34 and the width is 8.

    Exercise \(\PageIndex{44}\)

    The measure of one of the small angles of a right triangle is 14 more than 3 times the measure of the other small angle. Find the measure of both angles.

    Exercise \(\PageIndex{45}\)

    The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.

    Answer

    \(\text { The measures are } 16^{\circ} \text { and } 74^{\circ}\)

    Exercise \(\PageIndex{46}\)

    The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.

    Exercise \(\PageIndex{47}\)

    The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.

    Answer

    The measures are \(45^{\circ}\) and \(45^{\circ} .\)

    Exercise \(\PageIndex{48}\)

    Maxim has been offered positions by two car dealers. The first company pays a salary of $10,000 plus a commission of $1,000 for each car sold. The second pays a salary of $20,000 plus a commission of $500 for each car sold. How many cars would need to be sold to make the total pay the same?

    Exercise \(\PageIndex{49}\)

    Jackie has been offered positions by two cable companies. The first company pays a salary of $ 14,000 plus a commission of $100 for each cable package sold. The second pays a salary of $20,000 plus a commission of $25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?

    Answer

    80 cable packages would need to be sold.

    Exercise \(\PageIndex{50}\)

    Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?

    Exercise \(\PageIndex{51}\)

    Mitchell currently sells stoves for company A at a salary of $12,000 plus a $150 commission for each stove he sells. Company B offers him a position with a salary of $24,000 plus a $50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?

    Answer

    Mitchell would need to sell 120 stoves.

    Everyday Math

    Exercise \(\PageIndex{52}\)

    When Gloria spent 15 minutes on the elliptical trainer and then did circuit training for 30 minutes, her fitness app says she burned 435 calories. When she spent 30 minutes on the elliptical trainer and 40 minutes circuit training she burned 690 calories. Solve the system \(\left\{\begin{array}{l}{15 e+30 c=435} \\ {30 e+40 c=690}\end{array}\right.\) for e, the number of calories she burns for each minute on the elliptical trainer, and cc, the number of calories she burns for each minute of circuit training.

    Exercise \(\PageIndex{53}\)

    Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of 56 miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. Solve the system \(\left\{\begin{array}{l}{56 s=70 t} \\ {s=t+\frac{1}{2}}\end{array}\right.\)

    1. for t to find out how long it will take Tina to catch up to Stephanie.
    2. what is the value of ss, the number of hours Stephanie will have driven before Tina catches up to her?
    Answer
    1. \(t=2\) hours
    2. \(s=2 \frac{1}{2}\) hours

    Writing Exercises

    Exercise \(\PageIndex{54}\)

    Solve the system of equations
    \(\left\{\begin{array}{l}{x+y=10} \\ {x-y=6}\end{array}\right.\)

    1. by graphing.
    2. by substitution.
    3. Which method do you prefer? Why?

    Exercise \(\PageIndex{55}\)

    Solve the system of equations
    \(\left\{\begin{array}{l}{3 x+y=12} \\ {x=y-8}\end{array}\right.\) by substitution and explain all your steps in words.

    Answer

    Answers will vary.

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This figure shows a table with three rows and four columns. The columns are labeled, “I can…,” “Confidently.” “With some help.” and “No - I don’t get it.” The only column with filled in cells below it is labeled “I can…” It reads, “solve a system of equations by substitution.” “solve applications of systems of equations by substitution.”

    b. After reviewing this checklist, what will you do to become confident for all objectives?