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Mathematics LibreTexts

5.3E: Exercises

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    30524
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    Practice Makes Perfect

    Solve a System of Equations by Elimination

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

    Exercise \(\PageIndex{1}\)

    \(\left\{\begin{array}{l}{5 x+2 y=2} \\ {-3 x-y=0}\end{array}\right.\)

    Exercise \(\PageIndex{2}\)

    \(\left\{\begin{array}{l}{-3 x+y=-9} \\ {x-2 y=-12}\end{array}\right.\)

    Answer

    \((6,9)\)

    Exercise \(\PageIndex{3}\)

    \(\left\{\begin{array}{l}{6 x-5 y=-1} \\ {2 x+y=13}\end{array}\right.\)

    Exercise \(\PageIndex{4}\)

    \(\left\{\begin{array}{l}{3 x-y=-7} \\ {4 x+2 y=-6}\end{array}\right.\)

    Answer

    \((-2,1)\)

    Exercise \(\PageIndex{5}\)

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

    Exercise \(\PageIndex{6}\)

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

    Answer

    \((-7,-1)\)

    Exercise \(\PageIndex{7}\)

    \(\left\{\begin{array}{l}{3 x-2 y=1} \\ {-x+2 y=9}\end{array}\right.\)

    Exercise \(\PageIndex{8}\)

    \(\left\{\begin{array}{l}{-7 x+6 y=-10} \\ {x-6 y=22}\end{array}\right.\)

    Answer

    \((-2,-4)\)

    Exercise \(\PageIndex{9}\)

    \(\left\{\begin{array}{l}{3 x+2 y=-3} \\ {-x-2 y=-19}\end{array}\right.\)

    Exercise \(\PageIndex{10}\)

    \(\left\{\begin{array}{l}{5 x+2 y=1} \\ {-5 x-4 y=-7}\end{array}\right.\)

    Answer

    \((-1,3)\)

    Exercise \(\PageIndex{11}\)

    \(\left\{\begin{array}{l}{6 x+4 y=-4} \\ {-6 x-5 y=8}\end{array}\right.\)

    Exercise \(\PageIndex{12}\)

    \(\left\{\begin{array}{l}{3 x-4 y=-11} \\ {x-2 y=-5}\end{array}\right.\)

    Answer

    \((-1,2)\)

    Exercise \(\PageIndex{13}\)

    \(\left\{\begin{array}{l}{5 x-7 y=29} \\ {x+3 y=-3}\end{array}\right.\)

    Exercise \(\PageIndex{14}\)

    \(\left\{\begin{array}{l}{6 x-5 y=-75} \\ {-x-2 y=-13}\end{array}\right.\)

    Answer

    \((-5,9)\)

    Exercise \(\PageIndex{15}\)

    \(\left\{\begin{array}{l}{-x+4 y=8} \\ {3 x+5 y=10}\end{array}\right.\)

    Exercise \(\PageIndex{16}\)

    \(\left\{\begin{array}{l}{2 x-5 y=7} \\ {3 x-y=17}\end{array}\right.\)

    Answer

    \((6,1)\)

    Exercise \(\PageIndex{17}\)

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

    Exercise \(\PageIndex{18}\)

    \(\left\{\begin{array}{l}{7 x+y=-4} \\ {13 x+3 y=4}\end{array}\right.\)

    Answer

    \((-2,10)\)

    Exercise \(\PageIndex{19}\)

    \(\left\{\begin{array}{l}{-3 x+5 y=-13} \\ {2 x+y=-26}\end{array}\right.\)

    Exercise \(\PageIndex{20}\)

    \(\left\{\begin{array}{l}{3 x-5 y=-9} \\ {5 x+2 y=16}\end{array}\right.\)

    Answer

    \((2,3)\)

    Exercise \(\PageIndex{21}\)

    \(\left\{\begin{array}{l}{4 x-3 y=3} \\ {2 x+5 y=-31}\end{array}\right.\)

    Exercise \(\PageIndex{22}\)

    \(\left\{\begin{array}{l}{4 x+7 y=14} \\ {-2 x+3 y=32}\end{array}\right.\)

    Answer

    \((-7,6)\)

    Exercise \(\PageIndex{23}\)

    \(\left\{\begin{array}{l}{5 x+2 y=21} \\ {7 x-4 y=9}\end{array}\right.\)

    Exercise \(\PageIndex{24}\)

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

    Answer

    \((-9,3)\)

    Exercise \(\PageIndex{25}\)

    \(\left\{\begin{array}{l}{11 x+9 y=-5} \\ {7 x+5 y=-1}\end{array}\right.\)

    Exercise \(\PageIndex{26}\)

    \(\left\{\begin{array}{l}{3 x+8 y=67} \\ {5 x+3 y=60}\end{array}\right.\)

    Answer

    \((9,5)\)

    Exercise \(\PageIndex{27}\)

    \(\left\{\begin{array}{l}{2 x+9 y=-4} \\ {3 x+13 y=-7}\end{array}\right.\)

    Exercise \(\PageIndex{28}\)

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

    Answer

    \((-3,2)\)

    Exercise \(\PageIndex{29}\)

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

    Exercise \(\PageIndex{30}\)

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

    Answer

    \((-2,3)\)

    Exercise \(\PageIndex{31}\)

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

    Exercise \(\PageIndex{32}\)

    \(\left\{\begin{array}{l}{2 x+y=3} \\ {6 x+3 y=9}\end{array}\right.\)

    Answer

    infinitely many solutions

    Exercise \(\PageIndex{33}\)

    \(\left\{\begin{array}{l}{x-4 y=-1} \\ {-3 x+12 y=3}\end{array}\right.\)

    Exercise \(\PageIndex{34}\)

    \(\left\{\begin{array}{l}{-3 x-y=8} \\ {6 x+2 y=-16}\end{array}\right.\)

    Answer

    infinitely many solutions

    Exercise \(\PageIndex{35}\)

    \(\left\{\begin{array}{l}{4 x+3 y=2} \\ {20 x+15 y=10}\end{array}\right.\)

    Exercise \(\PageIndex{36}\)

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

    Answer

    infinitely many solutions

    Exercise \(\PageIndex{37}\)

    \(\left\{\begin{array}{l}{5 x-8 y=12} \\ {10 x-16 y=20}\end{array}\right.\)

    Exercise \(\PageIndex{38}\)

    \(\left\{\begin{array}{l}{-11 x+12 y=60} \\ {-22 x+24 y=90}\end{array}\right.\)

    Answer

    inconsistent, no solution

    Exercise \(\PageIndex{39}\)

    \(\left\{\begin{array}{l}{7 x-9 y=16} \\ {-21 x+27 y=-24}\end{array}\right.\)

    Exercise \(\PageIndex{40}\)

    \(\left\{\begin{array}{l}{5 x-3 y=15} \\ {y=\frac{5}{3} x-2}\end{array}\right.\)

    Answer

    inconsistent, no solution

    Exercise \(\PageIndex{41}\)

    \(\left\{\begin{array}{l}{2 x+4 y=7} \\ {y=-\frac{1}{2} x-4}\end{array}\right.\)

    Solve Applications of Systems of Equations by Elimination

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

    Exercise \(\PageIndex{42}\)

    The sum of two numbers is 65. Their difference is 25. Find the numbers.

    Answer

    The numbers are 20 and 45.

    Exercise \(\PageIndex{43}\)

    The sum of two numbers is 37. Their difference is 9. Find the numbers.

    Exercise \(\PageIndex{44}\)

    The sum of two numbers is −27. Their difference is −59. Find the numbers.

    Answer

    The numbers are 16 and −43.

    Exercise \(\PageIndex{45}\)

    The sum of two numbers is −45. Their difference is −89. Find the numbers.

    Exercise \(\PageIndex{46}\)

    Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for $114 or she is able to buy 2 shirts and 4 sweaters for $164. How much does a shirt cost? How much does a sweater cost?

    Answer

    A shirt costs $16 and a sweater costs $33.

    Exercise \(\PageIndex{47}\)

    Peter is buying office supplies. He is able to buy 3 packages of paper and 4 staplers for $40 or he is able to buy 5 packages of paper and 6 staplers for $62. How much does a package of paper cost? How much does a stapler cost?

    Exercise \(\PageIndex{48}\)

    The total amount of sodium in 2 hot dogs and 3 cups of cottage cheese is 4720 mg. The total amount of sodium in 5 hot dogs and 2 cups of cottage cheese is 6300 mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?

    Answer

    There are 860 mg in a hot dog. There are 1,000 mg in a cup of cottage cheese.

    Exercise \(\PageIndex{49}\)

    The total number of calories in 2 hot dogs and 3 cups of cottage cheese is 960 calories. The total number of calories in 5 hot dogs and 2 cups of cottage cheese is 1190 calories. How many calories are in a hot dog? How many calories are in a cup of cottage cheese?

    Choose the Most Convenient Method to Solve a System of Linear Equations

    In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

    Exercise \(\PageIndex{50}\)

    1. \( \left\{\begin{array}{l}{8 x-15 y=-32} \\ {6 x+3 y=-5}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{x=4 y-3} \\ {4 x-2 y=-6}\end{array}\right.\)
    Answer
    1. elimination
    2. substitution

    Exercise \(\PageIndex{51}\)

    1. \(\left\{\begin{array}{l}{y=7 x-5} \\ {3 x-2 y=16}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{12 x-5 y=-42} \\ {3 x+7 y=-15}\end{array}\right.\)

    Exercise \(\PageIndex{52}\)

    1. \(\left\{\begin{array}{l}{y=4 x+9} \\ {5 x-2 y=-21}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{9 x-4 y=24} \\ {3 x+5 y=-14}\end{array}\right.\)
    Answer
    1. substitution
    2. elimination

    Exercise \(\PageIndex{53}\)

    1. \(\left\{\begin{array}{l}{14 x-15 y=-30} \\ {7 x+2 y=10}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{x=9 y-11} \\ {2 x-7 y=-27}\end{array}\right.\)

    Everyday Math

    Exercise \(\PageIndex{54}\)

    In one hour Norris can row 3 miles upstream against the current. In the same amount of time he can row 5 miles downstream, with the current. Solve the system. \(\left\{\begin{array}{l}{r-c=3} \\ {r+c=5}\end{array}\right.\)

    1. for r, his rowing speed in still water.
    2. Then solve for c, the speed of the river current.
    Answer
    1. r=4
    2. c=1

    Exercise \(\PageIndex{55}\)

    Josie wants to make 10 pounds of trail mix using nuts and raisins, and she wants the total cost of the trail mix to be $54. Nuts cost $6 per pound and raisins cost $3 per pound. Solve the system \(\left\{\begin{array}{l}{n+r=10} \\ {6 n+3 r=54}\end{array}\right.\) to find n, the number of pounds of nuts, and rr, the number of pounds of raisins she should use.

    Writing Exercises

    Exercise \(\PageIndex{56}\)

    Solve the system
    \(\left\{\begin{array}{l}{x+y=10} \\ {5 x+8 y=56}\end{array}\right.\)

    1. by substitution
    2. by graphing
    3. Which method do you prefer? Why?
    Answer
    1. (8, 2)

    This image is a graph that shows the solution to the system “x plus y equals 10” and 5x plus 8y equals 56. The solution is on an x, y coordinate plane. Two arrows intersect at points 8 and 2.

    3. Answers will vary.

    Exercise \(\PageIndex{57}\)

    Solve the system \(\left\{\begin{array}{l}{x+y=-12} \\ {y=4-\frac{1}{2} x}\end{array}\right.\)

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

    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 four 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 elimination.” “solve applications of systems of equations by elimination.” and “choose the most convenient method to solve a system of linear equations.”

    b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?