3.5: Chapter 3 Review

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Systems of Equations

For the exercises 1-4, determine whether the given ordered pair is a solution to the system of equations.

1) $\begin{align*} -3x-5y &= 13\\ -x+4y &= 10 \end{align*}\; \text{ and } (-6,1)$

Answer: Yes

2) $\begin{align*} 3x+7y &= 1\\ 2x+4y &= 0 \end{align*}\; \text{ and } (2,3)$

3) $\begin{align*} 6x-y+3z &= 6\\ 3x+5y+2z &= 0\\ x+y &= 0 \end{align*}\; \; \text{ and }\; (3,-3,-5)$

Answer: No

4) $\begin{align*} 6x-7y+z &= 2\\ -x-y+3z &= 4\\ 2x+y-z &= 1 \end{align*}\; \; \text{ and }\; (4,2,-6)$

For the exercises 5-6, solve each system by graphing.

5) $\begin{align*} y &= 2x-7\\ y &= -x+2 \end{align*}$

Answer: $(3,-1)$

6) $\begin{align*} y &= -\dfrac{1}{2}x-1\\ y &= 3x+6\end{align*}$

For the exercises 7-8, solve each system by substitution.

7) $\begin{align*} x+5y &= 5\\ 2x+3y &= 4 \end{align*}$

Answer: $(-1,2)$

8) $\begin{align*} x-0.2y &= 1\\ -10x+2y &= 5 \end{align*}$

For the exercises 9-10, solve each system by elimination by addition.

9) $\begin{align*} 5x-2y &= 10\\ -3x+2y &= -2 \end{align*}$

Answer: $(4, 5)$

10) $\begin{align*} -x+2y &= -1\\ 5x-10y &= 6 \end{align*}$

Matrices and Matrix Operations

For the exercises 11-14, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

\[A=\begin{bmatrix} 1 & 3\\ 0 & 7 \end{bmatrix}, B=\begin{bmatrix} 2 & 14\\ 22 & 6 \end{bmatrix}, C=\begin{bmatrix} 1 & 5\\ 8 & 92\\ 12 & 6 \end{bmatrix}, D=\begin{bmatrix} 10 & 14\\ 7 & 2\\ 5 & 61 \end{bmatrix}, E=\begin{bmatrix} 6 & 12\\ 14 & 5 \end{bmatrix}, F=\begin{bmatrix} 0 & 9\\ 78 & 17\\ 15 & 4 \end{bmatrix} \nonumber\]

11) $C+D$

Answer: $\begin{bmatrix} 11 & 19\\ 15 & 94\\ 17 & 67 \end{bmatrix}$

12) $A+C$

13) $B-E$

Answer: $\begin{bmatrix} -4 & 2\\ 8 & 1 \end{bmatrix}$

14) $C+F$

For the exercises 15-18, use the matrices below to perform scalar multiplication.

\[A=\begin{bmatrix} 4 & 6\\ 13 & 12 \end{bmatrix}, B=\begin{bmatrix} 3 & 9\\ 21 & 12\\ 0 & 64 \end{bmatrix}, C=\begin{bmatrix} 16 & 3 & 7 & 18\\ 90 & 5 & 3 & 29 \end{bmatrix} \nonumber\]

15) $3B$

Answer: $\begin{bmatrix} 9 & 27\\ 63 & 36\\ 0 & 192 \end{bmatrix}$

16) $-2A$

17) $-4C$

Answer: $\begin{bmatrix} -64 & -12 & -28 & -72\\ -360 & -20 & -12 & -116 \end{bmatrix}$

18) $\dfrac{1}{2}C$

For the exercises 19-20, use the matrices below to perform matrix multiplication.

\[A=\begin{bmatrix} -1 & 5\\ 3 & 2 \end{bmatrix}, B=\begin{bmatrix} 3 & 6 & 4\\ -8 & 0 & 12 \end{bmatrix}, C=\begin{bmatrix} 4 & 10\\ -2 & 6\\ 5 & 9 \end{bmatrix} \nonumber\]

19) $BC$

Answer: $\begin{bmatrix} 20 & 102\\ 28 & 28 \end{bmatrix}$

20) $CA$

For the exercises 21-24, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.

\[A=\begin{bmatrix} 2 & -5\\ 6 & 7 \end{bmatrix}, B=\begin{bmatrix} -9 & 6\\ -4 & 2 \end{bmatrix}, C=\begin{bmatrix} 0 & 9\\ 7 & 1 \end{bmatrix}, D=\begin{bmatrix} -8 & 7 & -5\\ 4 & 3 & 2\\ 0 & 9 & 2 \end{bmatrix}, E=\begin{bmatrix} 4 & 5 & 3\\ 7 & -6 & -5\\ 1 & 0 & 9 \end{bmatrix} \nonumber\]

21) $4A+5D$

Answer: Undefined; dimensions do not match.

22) $2C+B$

23) $3D+4E$

Answer: $\begin{bmatrix} -8 & 41 & -3\\ 40 & -15 & -14\\ 4 & 27 & 42 \end{bmatrix}$

24) $C-0.5D$

Solving Systems with Gaussian Elimination

For the exercises 25-28, write the augmented matrix for the linear system.

25) $\begin{align*} 8x-37y &= 8\\ 2x+12y &= 3 \end{align*}$

26) $\begin{align*} 16y &= 4\\ 9x-y &= 2 \end{align*}$

Answer: $\left [ \begin{array}{cc|c} 0 & 16 & 4\\ 9 & -1 & 2\\ \end{array} \right ]$

27) $\begin{align*} 3x+2y+10z &= 3\\ -6x+2y+5z &= 13\\ 4x+z &= 18 \end{align*}$

28) $\begin{align*} x+5y+8z &= 19\\ 12x+3y &= 4\\ 3x+4y+9z &= -7 \end{align*}$

Answer: $\left [ \begin{array}{ccc|c} 1 & 5 & 8 & 16\\ 12 & 3 & 0 & 4\\ 3 & 4 & 9 & -7\end{array} \right ]$

For the exercises 29-32, write the linear system from the augmented matrix.

29) $\left [ \begin{array}{cc|c} -2 & 5 & 5\\ 6 & -18 & 26\\ \end{array} \right ]$

Answer: $\begin{align*} -2x+5y &= 5\\ 6x-18y &= 26 \end{align*}$

30) $\left [ \begin{array}{cc|c} 3 & 4 & 10\\ 10 & 17 & 439\\ \end{array} \right ]$

31) $\left [ \begin{array}{ccc|c} 3 & 2 & 0 & 3\\ -1 & -9 & 4& -1\\ 8 & 5 & 7 & 8\\ \end{array} \right ]$

Answer: $\begin{align*} 3x+2y &= 13\\ -x-9y+4z &= 53\\ 8x+5y+7z &= 80 \end{align*}$

32) $\left [ \begin{array}{ccc|c} 8 & 29 & 1 & 43\\ -1 & 7 & 5 & 38\\ 0 & 0 & 3 & 10\\ \end{array} \right ]$

For the exercises 33-38, solve the system by Gaussian elimination.

33) $\begin{align*} 2x-3y &= -9\\ 5x+4y &= 58 \end{align*}$

Answer: $(6,7)$

34) $\begin{align*} 6x+2y &= -4\\ 3x+4y &= -17 \end{align*}$

35) $\begin{align*} -60x+45y &= 12\\ 20x-15y &= -4 \end{align*}$

Answer: $\left (x, \dfrac{4}{15}(5x+1) \right )$

36) $\begin{align*} 0.5x+0.2y-0.3z &= 1\\ 0.4x-0.6y+0.7z &= 0.8\\ 0.3x-0.1y-0.9z &= 0.6 \end{align*}$

37) $\begin{align*} \dfrac{4}{5}x-\dfrac{7}{8}y+\dfrac{1}{2}z &= 1\\ -\dfrac{4}{5}x-\dfrac{3}{4}y+\dfrac{1}{3}z &= -8\\ -\dfrac{2}{5}x-\dfrac{7}{8}y+\dfrac{1}{2}z &= -5 \end{align*}$

Answer: $(5,12,15)$

38) $\begin{align*} x+y+z &= 14\\ 2y+3z &= -14\\ -16y-24z &= -112 \end{align*}$

Solving Systems with Inverses

In the exercises 39-40, show that matrix $A$ is the inverse of matrix $B$.

39) $A = \begin{bmatrix} 1 & 0\\ -1 & 1 \end{bmatrix}, B = \begin{bmatrix} 1 & 0\\ 1 & 1 \end{bmatrix}$

Answer: $AB = BA = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} = I$

40) $A = \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}, B = \begin{bmatrix} -2 & 1\\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}$

For the exercises 41-44, find the multiplicative inverse of each matrix, if it exists.

41) $\begin{bmatrix} 3 & -2\\ 1 & 9 \end{bmatrix}$

Answer: $\dfrac{1}{29}\begin{bmatrix} \dfrac{9}{29} & \dfrac{2}{29}\\ \dfrac{-1}{29} & \dfrac{3}{29} \end{bmatrix}$

42) $\begin{bmatrix} 1 & 1\\ 2 & 2 \end{bmatrix}$

43) $\begin{bmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2}\\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5}\\ \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \end{bmatrix}$

Answer: $\begin{bmatrix} 18 & 60 & -168\\ -56 & -140 & 448\\ 40 & 80 & -280 \end{bmatrix}$

44) $\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}$

For the exercises 45-48, solve the system using the inverse of a matrix.

45) $\begin{align*} 5x-6y &= -61\\ 4x+3y &= -2 \end{align*}$

Answer: $(-5,6)$

46) $\begin{align*} 8x+4y &= -100\\ 3x-4y &= 1 \end{align*}$

47) $\begin{align*} 3x-2y+5z &= 21\\ 5x+4y &= 37\\ x-2y-5z &= 5 \end{align*}$

Answer: $\left (7, \dfrac{1}{2}, \dfrac{1}{5} \right )$

48) $\begin{align*} 4x+4y+4z &= 40\\ 2x-3y+4z &= -12\\ -x+3y+4z &= 9 \end{align*}$

Real-World Applications

For the exercises 49-63, write a system of equations to represent the scenario. Then use any method to solve for the desired quantity.

49) A fast-food restaurant has a cost of production $C(x)=11x+120$ and a revenue function $R(x)=5x$. When does the company start to turn a profit?

Answer: They never turn a profit.

50) A cell phone factory has a cost of productiona $C(x)=150x+10,000$ and a revenue function $R(x)=200x$. What is the break-even point?

51) A musician charges $C(x)=64x+20,000$, where $x$ is the total number of attendees at the concert. The venue charges $\$80$ per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

Answer: $(1,250, 100,000)$

52) A guitar factory has a cost of production $C(x)=75x+50,000$. If the company needs to break even after $150$ units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.

53) The startup cost for a restaurant is $\$120,000$, and each meal costs $\$10$ for the restaurant to make. If each meal is then sold for $\$15$, after how many meals does the restaurant break even?

Answer: $24,000$

54) A moving company charges a flat rate of $\$150$, and an additional $\$5$ for each box. If a taxi service would charge $\$20$ for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?

55) There were $130$ faculty at a conference. If there were $18$ more women than men attending, how many of each gender attended the conference?

Answer: $56$ men, $74$ women

56) An investor earned triple the profits of what she earned last year. If she made $\$500,000.48$ total for both years, how much did she earn in profits each year?

57) An investor who dabbles in real estate invested $1.1$ million dollars into two land investments. On the first investment, Swan Peak, her return was a $110\%$ increase on the money she invested. On the second investment, Riverside Community, she earned $50\%$ over what she invested. If she earned $\$1$ million in profits, how much did she invest in each of the land deals?

Answer: Swan Peak: $\$750,000$, Riverside: $\$350,000$

58) If an investor invests a total of $\$25,000$ into two bonds, one that pays $3\%$ simple interest, and the other that pays $2\dfrac{7}{8}\%$ interest, and the investor earns $\$737.50$ annual interest, how much was invested in each account?

59) If an investor invests $\$23,000$ into two bonds, one that pays $4\%$ in simple interest, and the other paying $2\%$ simple interest, and the investor earns $\$710.00$ annual interest, how much was invested in each account?

Answer: $\$12,500$ in the first account, $\$10,500$ in the second account.

60) A concert manager counted $350$ ticket receipts the day after a concert. The price for a student ticket was $\$12.50$, and the price for an adult ticket was $\$16.00$. The register confirms that $\$5,075$ was taken in. How many student tickets and adult tickets were sold?

61) A local band sells out for their concert. They sell all $1,175$ tickets for a total purse of $\$28,112.50$. The tickets were priced at $\$20$ for student tickets, $\$22.50$ for children, and $\$29$ for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?

Answer: $500$ students, $225$ children, and $450$ adults

62) You invested $\$10,000$ into two accounts: one that has simple $3\%$ interest, the other with $2.5\%$ interest. If your total interest payment after one year was $\$283.50$, how much was in each account after the year passed?

63) You invested $\$2,300$ into account 1, and $\$2,700$ into account 2. If the total amount of interest after one year is $\$254$, and account 2 has $1.5$ times the interest rate of account 1, what are the interest rates? Assume simple interest rates.

Answer: $4\%$ for account 1, $6\%$ for account 2

Contributors and Attributions

Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus.