4.E: Systems of Linear Equations (Exercises)

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4.1: Solving Systems by Graphing

In Exercises 1-6, solve each of the given systems by sketching the lines represented by each equation in the system, then determining the coordinates of the point of intersection. Each of these problems have been designed so that the coordinates of the intersection point are integers. Check your solution.

1) $\begin{aligned} 3 x-4 y &=24 \\ y &=-\dfrac{1}{2} x-1 \end{aligned}$

Answer: $(4,-3)$

2) $\begin{aligned} x-4 y &=-8 \\ y &=\dfrac{5}{4} x+6 \end{aligned}$

3) $\begin{aligned} 2 x+y &=6 \\ y &=x+3 \end{aligned}$

Answer: $(1,4)$

4) $\begin{aligned} x-2 y &=-4 \\ y &=\dfrac{5}{2} x+6 \end{aligned}$

5) $\begin{aligned} x+2 y &=-6 \\ y &=-3 x-8 \end{aligned}$

Answer: $(-2,-2)$

6) $\begin{aligned} x-3 y &=6 \\ y &=2 x-7 \end{aligned}$

In Exercises 7-18, solve each of the given systems by sketching the lines represented by each equation of the given system on graph paper, then estimating the coordinates of the point of intersection to the nearest tenth. Check the solution.

7) $\begin{aligned}x-3 y&=3 \\ x-4 y&=-4\end{aligned}$

Answer: $(-3.4,0.1)$

8) $\begin{aligned}4 x-3 y&=-12 \\ -x-4 y&=4\end{aligned}$

9) $\begin{aligned}-3 x+3 y&=-9 \\ -3 x+3 y&=-12\end{aligned}$

Answer: No solution. Lines are parallel.

10) $\begin{aligned} x-y &=-2 \\ 2 x-2 y &=6 \end{aligned}$

11) $\begin{aligned} 6 x-7 y &=-42 \\ y &=-\dfrac{1}{4} x+4 \end{aligned}$

Answer: $(-1.8,4.5)$

12) $\begin{aligned} 4 x+3 y &=24 \\ y &=\dfrac{1}{7} x+5 \end{aligned}$

13) $\begin{aligned} 6 x-7 y &=-42 \\ y &=-\dfrac{1}{5} x+2 \end{aligned}$

Answer: $(-3.8,2.8)$

14) $\begin{aligned} 7 x-8 y &=56 \\ y &=-\dfrac{1}{3} x-4 \end{aligned}$

15) $\begin{aligned}6 x+3 y&=12 \\ -2 x-y&=4 \end{aligned}$

Answer: No solution. Lines are parallel.

16) $\begin{aligned} x-4 y &=-4 \\-x+4 y &=-4 \end{aligned}$

17) $\begin{aligned}3 x+y&=3 \\ -2 x+3 y&=-6\end{aligned}$

Answer: $(1.4,-1.1)$

18) $\begin{aligned}2 x-y&=-2 \\ -x-2 y&=4\end{aligned}$

In Exercises 19-24, use the graphing calculator to solve the given system. Round your answer to the nearest tenth. Use the Calculator Submission Guidelines from Chapter 3, Section 2, when reporting your answer on your homework.

19) $\begin{aligned}y&=\dfrac{3}{4} x+7 \\ y&=-\dfrac{1}{3} x+2 \end{aligned}$

Answer: $(-4.6,3.5)$

20) $\begin{aligned}y&=\dfrac{7}{6} x+6 \\ y&=-\dfrac{1}{7} x+3\end{aligned}$

21) $\begin{aligned}y&=\dfrac{4}{3} x-3 \\ y&=-\dfrac{4}{7} x-1\end{aligned}$

Answer: $(1.1,-1.6)$

22) $\begin{aligned}y&=\dfrac{8}{3} x-3 \\ y&=-\dfrac{1}{8} x-2\end{aligned}$

23) $\begin{aligned}y&=\dfrac{1}{6} x+1 \\ y&=-\dfrac{3}{7} x+5\end{aligned}$

Answer: $(6.7,2.1)$

24) $\begin{aligned}y&=\dfrac{5}{8} x+3 \\ y&=-\dfrac{5}{6} x-5\end{aligned}$

In Exercises 25-30, use the graphing calculator to solve the given system. Round your answer to the nearest tenth. Use the Calculator Submission Guidelines when reporting your answer on your homework.

25) $\begin{array}{c}{6 x+16 y=96} \\ {-6 x+13 y=-78}\end{array}$

Answer: $(14.3,0.6)$

26) $\begin{aligned}-4 x+16 y &=-64 \\ 5 x+8 y &=40 \end{aligned}$

27) $\begin{aligned}-2 x-11 y &=22 \\ 8 x-12 y &=-96 \end{aligned}$

Answer: $(-11.8,0.1)$

28) $\begin{aligned}-6 x-10 y &=60 \\ 2 x-18 y &=-36 \end{aligned}$

29) $\begin{array}{c}{-6 x+2 y=-12} \\ {-12 x+3 y=-36}\end{array}$

Answer: $(6.0,12.0)$

30) $\begin{aligned}-3 x+y &=-3 \\-14 x+3 y &=-42 \end{aligned}$

4.2: Solving Systems by Substitution

In Exercises 1-8, use the substitution method to solve each of the following systems. Check your answer manually, without the use of a calculator.

1) $\begin{aligned}-7 x+7 y &=63 \\ y &=6-2 x \end{aligned}$

Answer: $(-1,8)$

2) $\begin{aligned} 3 x-8 y &=27 \\ y &=4-7 x \end{aligned}$

3) $\begin{aligned} x &=19+7 y \\ 3 x-3 y &=3 \end{aligned}$

Answer: $(3,-4)$

4) $\begin{aligned} x &=39+8 y \\-9 x+2 y &=-71 \end{aligned}$

5) $\begin{aligned} x &=-5-2 y \\-2 x-6 y &=18 \end{aligned}$

Answer: $(3,-4)$

6) $\begin{aligned} x &=15+6 y \\ 9 x+3 y &=21 \end{aligned}$

7) $\begin{aligned} 6 x-8 y &=24 \\ y &=15+3 x \end{aligned}$

Answer: $(-8,-9)$

8) $\begin{aligned} 9 x+8 y &=-45 \\ y &=15-8 x \end{aligned}$

In Exercises 9-28, use the substitution method to solve each of the following systems.

9) $\begin{aligned}-x+9 y&=46 \\ 7 x-4 y&=-27\end{aligned}$

Answer: $(-1,5)$

10) $\begin{aligned}-x+9 y&=-12 \\ 4 x+7 y&=-38\end{aligned}$

11) $\begin{aligned}-x+4 y&=22 \\ 8 x+7 y&=-20\end{aligned}$

Answer: $(-6,4)$

12) $\begin{aligned}-x-2 y&=15 \\ 3 x-9 y&=15 \end{aligned}$

13) $\begin{aligned}x+2 y&=-4 \\ 6 x-4 y&=-56 \end{aligned}$

Answer: $(-8,2)$

14) $\begin{aligned}x+8 y&=79 \\ 4 x+6 y&=8 \end{aligned}$

15) $\begin{aligned} x+6 y &=-49 \\-3 x+4 y &=-7 \end{aligned}$

Answer: $(-7,-7)$

16) $\begin{aligned} x-4 y &=33 \\ 4 x+7 y &=-6 \end{aligned}$

17) $\begin{aligned}-2 x+8 y&=-50 \\ -9 x-y&=-3 \end{aligned}$

Answer: $(1,-6)$

18) $\begin{aligned}-6 x-6 y &=102 \\ 9 x-y &=-63 \end{aligned}$

19) $\begin{aligned}-4 x-8 y &=-4 \\ y &=-2 y-4 \end{aligned}$

Answer: No solution

20) $\begin{aligned} 3 x+6 y &=2 \\ y &=-2 y+2 \end{aligned}$

21) $\begin{aligned}-2 x-2 y &=26 \\-7 x+y &=19 \end{aligned}$

Answer: $(-4,-9)$

22) $\begin{aligned}-2 x-8 y&=-30 \\ -6 x+y&=10 \end{aligned}$

23) $\begin{aligned}3 x-4 y&=-43 \\ -3 x+y&=22 \end{aligned}$

Answer: $(-5,7)$

24) $\begin{aligned}-2 x+8 y &=14 \\ 8 x+y &=43 \end{aligned}$

25) $\begin{aligned}-8 x-4 y &=24 \\ 9 x-y &=-71 \end{aligned}$

Answer: $(-7,8)$

26) $\begin{aligned}-8 x-2 y&=-14 \\ -6 x-y&=-9 \end{aligned}$

27) $\begin{aligned}-8 x-7 y &=2 \\ y &=-\dfrac{8}{7} x+9 \end{aligned}$

Answer: No solution

28) $\begin{aligned} 9 x+4 y &=-3 \\ y &=-\dfrac{9}{4} x+6 \end{aligned}$

In Exercises 29-36, use the substitution method to solve each of the following systems. Use your graphing calculator to check your solution.

29) $\begin{aligned}3 x-5 y&=3 \\ 5 x-7 y&=2 \end{aligned}$

Answer: $(-11 / 4,-9 / 4)$

30) $\begin{aligned}4 x-5 y&=4 \\ 3 x-2 y&=-1 \end{aligned}$

31) $\begin{aligned}4 x+3 y&=8 \\ 3 x+4 y&=2 \end{aligned}$

Answer: $(26 / 7,-16 / 7)$

32) $\begin{aligned}3 x+8 y&=3 \\ -4 x-9 y&=-2 \end{aligned}$

33) $\begin{aligned}3 x+8 y&=6 \\ 2 x+7 y&=-2 \end{aligned}$

Answer: $(58 / 5,-18 / 5)$

34) $\begin{aligned}3 x-7 y&=6 \\ 2 x-3 y&=1 \end{aligned}$

35) $\begin{aligned}4 x+5 y&=4 \\ -3 x-2 y&=1 \end{aligned}$

Answer: $(-13 / 7,16 / 7)$

36) $\begin{aligned}5 x+4 y&=5 \\ 4 x+5 y&=2 \end{aligned}$

In Exercises 37-48, use the substitution method to determine how many solutions each of the following linear systems has.

37) $\begin{aligned}-9 x+6 y &=9 \\ y &=\dfrac{3}{2} x-8 \end{aligned}$

Answer: No solutions

38) $\begin{aligned} 3 x-5 y &=9 \\ y &=\dfrac{3}{5} x+6 \end{aligned}$

39) $\begin{aligned} y &=-2 x-16 \\-14 x-7 y &=112 \end{aligned}$

Answer: Inﬁnite number of solutions

40) $\begin{aligned} y &=-12 x+12 \\ 120 x+10 y &=120 \end{aligned}$

41) $\begin{aligned} x &=16-5 y \\-4 x+2 y &=24 \end{aligned}$

Answer: One solution

42) $\begin{aligned} x &=-18-4 y \\ 7 x-7 y &=49 \end{aligned}$

43) $\begin{aligned} y &=7 y+18 \\ 9 x-63 y &=162 \end{aligned}$

Answer: Inﬁnite number of solutions

44) $\begin{aligned} y &=4 y-9 \\-10 x+40 y &=90 \end{aligned}$

45) $\begin{aligned} x &=-2 y+3 \\ 4 x+8 y &=4 \end{aligned}$

Answer: No solutions

46) $\begin{aligned} x &=2 y+4 \\-3 x+6 y &=5 \end{aligned}$

47) $\begin{aligned}-9 x+4 y &=73 \\ y &=-3-2 x \end{aligned}$

Answer: One solution

48) $\begin{aligned} 6 x+9 y &=27 \\ y &=16-5 x \end{aligned}$

4.3: Solving Systems by Elimination

In Exercises 1-8, use the elimination method to solve each of the following systems. Check your result manually, without the assistance of a calculator.

1) $\begin{aligned}x+4 y&=0 \\ 9 x-7 y&=-43\end{aligned}$

Answer: $(-4,1)$

2) $\begin{aligned} x+6 y &=-53 \\ 5 x-9 y &=47 \end{aligned}$

3) $\begin{aligned}6 x+y&=8 \\ 4 x+2 y&=0 \end{aligned}$

Answer: $(2,-4)$

4) $\begin{aligned} 4 x+y &=18 \\-2 x+6 y &=-22 \end{aligned}$

5) $\begin{aligned}-8 x+y&=-56 \\ 4 x+3 y&=56 \end{aligned}$

Answer: $(8,8)$

6) $\begin{aligned}2 x+y=21 \\ 7 x+8 y&=87 \end{aligned}$

7) $\begin{aligned} x+8 y &=41 \\-5 x-9 y &=-50 \end{aligned}$

Answer: $(1,5)$

8) $\begin{aligned} x-4 y &=-31 \\-2 x-6 y &=-36 \end{aligned}$

In Exercises 9-16, use the elimination method to solve each of the following systems.

9) $\begin{aligned}-12 x+9 y&=0 \\ -6 x-4 y&=-34 \end{aligned}$

Answer: $(3,4)$

10) $\begin{aligned}-27 x-5 y&=148 \\ -9 x-3 y&=60 \end{aligned}$

11) $\begin{aligned}27 x-6 y&=-96 \\ -3 x-5 y&=22 \end{aligned}$

Answer: $(-4,-2)$

12) $\begin{aligned}-8 x+8 y&=-32 \\ 2 x-9 y&=15 \end{aligned}$

13) $\begin{aligned}2 x-6 y&=28 \\ -3 x+18 y&=-60 \end{aligned}$

Answer: $(8,-2)$

14) $\begin{aligned}-8 x-6 y&=96 \\ 4 x+30 y&=-156 \end{aligned}$

15) $\begin{aligned}-32 x+7 y&=-238 \\ 8 x-4 y&=64 \end{aligned}$

Answer: $(7,-2)$

16) $\begin{aligned}12 x+6 y&=30 \\ -2 x+7 y&=51 \end{aligned}$

In Exercises 17-24, use the elimination method to solve each of the following systems.

17) $\begin{aligned} 3 x-7 y &=-75 \\-2 x-2 y &=-10 \end{aligned}$

Answer: $(-4,9)$

18) $\begin{aligned}-8 x+3 y&=42 \\ -7 x+8 y&=26 \end{aligned}$

19) $\begin{aligned}9 x-9 y&=-63 \\ 2 x-6 y&=-34 \end{aligned}$

Answer: $(-2,5)$

20) $\begin{aligned}-4 x-8 y&=-52 \\ -7 x-3 y&=-14 \end{aligned}$

21) $\begin{aligned}-9 x-2 y&=28 \\ 5 x-3 y&=-32 \end{aligned}$

Answer: $(-4,4)$

22) $\begin{aligned}-8 x-2 y&=-12 \\ 6 x+3 y&=12 \end{aligned}$

23) $\begin{aligned}-3 x-5 y &=-34 \\ 7 x+7 y &=56 \end{aligned}$

Answer: $(3,5)$

24) $\begin{aligned}-9 x-9 y&=9 \\ 7 x+4 y&=8 \end{aligned}$

In Exercises 25-32, use the elimination method to solve each of the following systems. Use your calculator to check your solutions.

25) $\begin{aligned}2 x-7 y&=-2 \\ 7 x+6 y&=3 \end{aligned}$

Answer: $(9 / 61,20 / 61)$

26) $\begin{aligned}-9 x-4 y &=4 \\ 5 x-3 y &=-1 \end{aligned}$

27) $\begin{aligned} 2 x+3 y &=-2 \\-5 x+5 y &=2 \end{aligned}$

Answer: $(-16 / 25,-6 / 25)$

28) $\begin{aligned}-5 x+8 y&=-3 \\ -4 x-7 y&=3 \end{aligned}$

29) $\begin{aligned}9 x+4 y&=-4 \\ -7 x-9 y&=3 \end{aligned}$

Answer: $(-24 / 53,1 / 53)$

30) $\begin{aligned}-3 x-5 y&=-4 \\ 4 x+6 y&=1 \end{aligned}$

31) $\begin{aligned}2 x+2 y&=4 \\ 3 x-5 y&=3 \end{aligned}$

Answer: $(13 / 8,3 / 8)$

32) $\begin{aligned}6 x-9 y&=-2 \\ -4 x-8 y&=4 \end{aligned}$

In Exercises 33-40, use the elimination method to determine how many solutions each of the following system of equations has.

33) $\begin{aligned} x+7 y &=-32 \\-8 x-56 y &=256 \end{aligned}$

Answer: Inﬁnite number of solutions

34) $\begin{aligned}-8 x+y&=-53 \\ 56 x-7 y&=371 \end{aligned}$

35) $\begin{aligned} 16 x-16 y &=-256 \\-8 x+8 y &=128 \end{aligned}$

Answer: Inﬁnite number of solutions

36) $\begin{aligned} 3 x-3 y &=42 \\-6 x+6 y &=-84 \end{aligned}$

37) $\begin{aligned}x-4 y&=-37 \\ 2 x-8 y&=54 \end{aligned}$

Answer: No solutions

38) $\begin{aligned}4 x+y&=-13 \\ 28 x+7 y&=189 \end{aligned}$

39) $\begin{aligned} x+9 y &=73 \\-4 x-5 y &=-44 \end{aligned}$

Answer: One solution

40) $\begin{aligned}6 x+y&=31 \\ -5 x-6 y&=-62 \end{aligned}$

4.4: Applications of Linear Systems

1) In geometry, two angles that sum to $90^{\circ}$ are called complementary angles. If the second of two complementary angles is $42$ degrees larger than $3$ times the ﬁrst angle, ﬁnd the degree measure of both angles.

Answer: $12^{\circ}$ and $78^{\circ}$

2) In geometry, two angles that sum to $90^{\circ}$ are called complementary angles. If the second of two complementary angles is $57$ degrees larger than $2$ times the ﬁrst angle, ﬁnd the degree measure of both angles.

3) The perimeter of a rectangle is $116$ inches. The length of the rectangle is $28$ inches more than twice the width. Find the width and length of the rectangle.

Answer: Length is $48$ inches, width is $10$ inches

4) The perimeter of a rectangle is $528$ inches. The length of the rectangle is $24$ inches more than twice the width. Find the width and length of the rectangle.

5) Maria has $\$6.35$ in change in her pocket, all in nickels and quarters. she has $59$ coins in all. How many quarters does she have?

Answer: $17$ quarters

6) Amy has $\$5.05$ in change in her pocket, all in nickels and quarters. she has $53$ coins in all. How many quarters does she have?

7) A store sells cashews for $\$6.00$ per pound and raisins for $\$7.00$ per pound. How many pounds of cashews and how many pounds of raisins should you mix to make a $50$-lb mixture costing $\$6.42$ per pound?

Answer: $29$ pounds of cashews, $21$ pounds of raisins

8) A store sells cashews for $\$3.00$ per pound and pecans for $\$8.00$ per pound. How many pounds of cashews and how many pounds of pecans should you mix to make a $50$-lb mixture costing $\$4.10$ per pound?

9) Roberto has $\$5.45$ in change in his pocket, all in dimes and quarters. He has $38$ coins in all. How many dimes does he have?

Answer: $27$ dimes

10) Benjamin has $\$7.40$ in change in his pocket, all in dimes and quarters. He has $44$ coins in all. How many dimes does he have?

11) In geometry, two angles that sum to $180^{\circ}$ are called supplementary angles. If the second of two supplementary angles is $40$ degrees larger than $3$ times the ﬁrst angle, ﬁnd the degree measure of both angles.

Answer: $35^{\circ}$ and $145^{\circ}$

12) In geometry, two angles that sum to $180^{\circ}$ are called supplementary angles. If the second of two supplementary angles is $114$ degrees larger than $2$ times the ﬁrst angle, ﬁnd the degree measure of both angles.

13) Eileen inherits $\$20,000$ and decides to invest the money in two accounts, part in a certiﬁcate of deposit that pays $3\%$ interest per year, and the rest in a mutual fund that pays $5\%$ per year. At the end of the ﬁrst year, her investments earn a total of $\$780$ in interest. Find the amount invested in each account.

Answer: $\$11,000$ in certiﬁcate of deposit, $\$9,000$ in mutual fund.

14) Alice inherits $\$40,000$ and decides to invest the money in two accounts, part in a certiﬁcate of deposit that pays $3\%$ interest per year, and the rest in a mutual fund that pays $6\%$ per year. At the end of the ﬁrst year, her investments earn a total of $\$1,980$ in interest. Find the amount invested in each account.

15) The perimeter of a rectangle is $376$ centimeters. The length of the rectangle is $12$ centimeters less than three times the width. Find the width and length of the rectangle.

Answer: Length is $138$ centimeters, width is $50$ centimeters

16) The perimeter of a rectangle is $344$ feet. The length of the rectangle is $28$ feet less than three times the width. Find the width and length of the rectangle.