# Virtual Math Learning Center

## Exam 1 Review

### Exercises

Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.

1. A line $$L$$ passes through the points $$(7, 2)$$ and $$(2, 5)$$. If $$x$$ increases by $$4$$, what is the change in $$y$$?

$$y$$ will decrease by $$\dfrac{12}{5}$$

If you would like to see more videos on this topic, click the following link and check the related videos.

2. Laura is planning to buy one 5-lb bag of sugar, two 5-lb bags of flour, four 1-gal cartons of milk, and one 1-dozen carton of large eggs. The prices of these items in three neighborhood supermarkets are as follows:

$$\begin{array}{l|cccc} & \text{Sugar} & \text{Flour} & \text{Milk} & \text{Eggs} \\ & \text{5-lb bag} & \text{5-lb bag} & \text{1-gal carton} & \text{1-doz carton} \\ \hline \text{Supermarket 1} & \3.29 & \3.29 & \2.79 & \2.99\\ \text{Supermarket 2} & \2.59 & \2.99 & \2.89 & \3.49 \\ \text{Supermarket 3} & \3.19 & \3.69 & \3.59 & \3.99 \\ \end{array}$$

1. Write a $$4 \times 3$$ matrix $$A$$ to represent the prices of items in the three supermarkets.
2. Write a row matrix $$B$$ to represent the quantities of the items that Laura plans to purchase in the three supermarkets.
3. Use matrix multiplication to find a matrix $$C$$ that represents Laura's total cost at each supermarket. Which supermarket should she buy from to minimize her cost? (Assuming she only shops at one supermarket)

1. $$A=\begin{array}{cc} & \begin{array}{ccc} \text{Market #1} & \text{Market #2} & \text{Market #3}\\ \end{array}\\ \begin{array}{l} \text{Sugar}\\ \text{Flour}\\ \text{Milk}\\ \text{Eggs} \end{array} & \left[\begin{array}{ccc} \hspace{1.5em}3.29\hspace{1.5em} & \hspace{1.5em}2.59\hspace{1.5em} & \hspace{1.5em}3.19\hspace{1.5em}\\ 3.29 & 2.99 & 3.69\\ 2.79 & 2.89 & 3.59\\ 2.99 & 3.49 & 3.99 \end{array}\right] \end{array}$$

2. $$\begin{array}{cc} & \begin{array}{cccc} \text{Sugar} & \text{Flour} & \text{Milk} & \text{Eggs}\\ \end{array}\\ \begin{array}{l} B=\text{Laura's Purchase} \end{array} & \left[\begin{array}{cccc} \hspace{1em}1\hspace{1em} & \hspace{1em}2\hspace{1em} & \hspace{1em}4\hspace{1em} & \hspace{1em}1\hspace{1em} \end{array}\right] \end{array}$$

3. $$\begin{array}{cc} & \begin{array}{ccc} \text{Market #1} & \text{Market #2} & \text{Market #3}\\ \end{array}\\ \begin{array}{l} C=BA=\text{Laura's Purchase} \end{array} & \left[\begin{array}{cccc} \hspace{1em}24.02\hspace{1em} & \hspace{1em}23.62\hspace{3em} & \hspace{1em}28.92\hspace{1em} \end{array}\right] \end{array}$$

Laura should shop at Market #2 because the cost will be the least for the number of each item she wants to purchase.

If you would like to see more videos on this topic, click the following link and check the related videos.

3. Melanie bought a brand-new car in 2008. In 2013 the car was worth $14,500 and in 2028 she sold it to the salvage yard (or scrap yard) for$5,500.  Assuming a linear depreciation model, find an equation that gives the value, $$V$$, of the car $$t$$ years after its initial purchase. Use the equation to determine the original amount that Melanie paid for the car in 2008.

$$V(t)=-600t+17500$$ for $$0 \leq t \leq 20$$ and  $$V(t)=5500$$ for $$t > 20$$. Melanie paid $17,500 for the car when she purchased it in 2008. If you would like to see more videos on this topic, click the following link and check the related videos. 4. A company makes and sells skateboards. The total profit (in dollars) of producing and selling $$x$$ skateboards is $$P(x) = 110x-17,050$$. The company sells the skateboards for \$200 each. Find the company's break-even point and interpret the meaning of this point.

Break-even point is $$(155, 31,000)$$.  This means the company must produce and sell 155 skateboards for a cost (or revenue) of $31,000. When they sell 155 skateboards, the profit is zero, so they need to sell more than 155 skateboards to start to earn a profit. If you would like to see more videos on this topic, click the following link and check the related videos. 5. Determine which of the following matrices below are in row reduced echelon form (RREF). $$\text{(a) }\left[\begin{array}{cccc|c} 1& 0&-3&0&0\$3pt] 0& 1&1&1&0\\[3pt] 0&0&0&0&1 \end{array}\right]\quad$$$$\text{(b) }\left[\begin{array}{ccc|c} 1&0&0&3\\[3pt] 0&0&1&6\\[3pt] 0&1&0&4\\[3pt] 0&0&0&0 \end{array}\right]\quad$$ $$\text{(c) }\left[\begin{array}{ccc|c} 1&0&2&3\\[3pt] 0&1&0&7\\[3pt] 0&0&1&9 \end{array}\right]\quad$$ 1. Yes 2. No 3. No If you would like to see more videos on this topic, click the following link and check the related videos. 6. Given the matrix equation below, find matrix $$D$$. \[2 \left[\begin{array}{ccc} 3 & 1 & -1\\[4pt] 0 & a & 4 \end{array}\right] \bullet \left[\begin{array}{ccc} 2 & -4 & x\\[4pt] 6 & 2 & -1 \end{array}\right]^T + \left[\begin{array}{cc} 1& -y\\[4pt] 3 &6 \end{array}\right]=D$ $$D=\left[\begin{array}{cc} 5-2x & 42-y\$4pt] 3-8a+8x & 4a-2 \end{array}\right]$$ If you would like to see more videos on this topic, click the following link and check the related videos. 7. Given matrix $$A$$ has dimensions $$3\times 4$$ and matrix $$B$$ has dimensions $$2\times3$$. Determine if $$BA^T$$ exists. If it does, give the dimensions of the matrix $$BA^T.$$ $$BA^T$$ is not defined. If you would like to see more videos on this topic, click the following link and check the related videos. 8. Given the dimensions of matrix $$E$$ are $$3\times 4$$, find the dimensions of matrix $$F$$ for which $$4E + F$$ is defined? $$3 \times 4$$ If you would like to see more videos on this topic, click the following link and check the related videos. 9. Write the correct augmented matrix for the system of linear equations given below. \[\begin{cases} -3y +z = -7\\[3pt] y=\dfrac{1}{2}x +4\\[3pt] 2x-2y +5z =-2 \end{cases}$ $$\left[\begin{array}{ccc|c} 0& -3& 1& -7\$3pt] -0.5& 1& 0& 4\\[3pt] 2& -2& 5& -2 \end{array}\right]$$ If you would like to see more videos on this topic, click the following link and check the related videos. 10. The parameterized solution to a system of linear equations with infinitely many solutions is given by $$(w,x,y,z)=(3 - 2t,\ \ 4,\ \ t,\ \ 1 - t)$$ where $$t$$ is any real number. Determine if the values below are a particular solution to the given system. $$\text{(a) }(w,x,y,z)=(3, 4, 0, 1) \quad$$ $$\text{(b) }(w,x,y,z)=(1, 4, 1, 1) \quad$$ $$\text{(c) }(w,x,y,z)=(5, 4, -1, 2)$$ 1. Yes 2. No 3. Yes If you would like to see more videos on this topic, click the following link and check the related videos. 11. Given a matrix below, which row operation must be performed to pivot about the entry in row one column one? $$\left[\begin{array}{ccc|c} 1& -2&-1& 3\;\\[3pt] 0& 6& 0& 1\\[3pt] -2& 0& 3& 2 \end{array}\right] \qquad \longrightarrow\qquad$$$$\left[\begin{array}{ccc|c} 1& -2&-1& 3\;\\[3pt] 0& 6& 0& 1\\[3pt] 0& -4& 1& 8 \end{array}\right]$$ $$2R_1+R_3 \longrightarrow R_3$$ If you would like to see more videos on this topic, click the following link and check the related videos. 12. Without the aid of technology, find the solution to the system of linear equations below. \[\begin{cases} -3x-6y=-12\\[3pt] -2x+3y=15\end{cases}$ $$\displaystyle \left( -\frac{18}{7}, \frac{23}{7} \right)$$ If you would like to see more videos on this topic, click the following link and check the related videos. 13. Without the aid of technology, find the solution to the system of linear equations given by $$y_1$$ (graphed below) and $$-6x+4y=0.$$ No Solution. If you would like to see more videos on this topic, click the following link and check the related videos. 14. Use technology to find the solution to the following system of linear equations.$\begin{cases} 2x+y -z =1\\[3pt] 3x+4y+2z=13\\[3pt] x-5y-2z =0 \end{cases}$ $$(x,y,z)=(3,-1,4)$$ If you would like to see more videos on this topic, click the following link and check the related videos. 15. Use technology to find the solution to the following system of linear equations.$\begin{cases} x+y +2z =-2\\[3pt] 3x-y+14z=6\\[3pt] x+2y =-5 \end{cases}$ $$(x,y,z)=(-4t+1, 2t-3, t)$$ If you would like to see more videos on this topic, click the following link and check the related videos. 16. Given matrices $$A$$ and $$B$$ below, find the products $$AB$$ and $$BA$$, if they exist.$A=\left[\begin{array}{cc} x&0\;\\[5pt] 2&4 \end{array}\right], \quad B= \left[\begin{array}{ccc} 3&-3&-2\;\\[5pt] 1&4&3 \end{array}\right]$ 1. $$AB=\left[\begin{array}{ccc} 3x & -3x & -2x\$4pt] 10 & 10 & 8 \end{array}\right]$$ 2. $$BA$$ is not possible. If you would like to see more videos on this topic, click the following link and check the related videos. 17. Each day you feed your dog a mixture of three kinds of food: Kibble, Bits, and Chunks. Matrix $$M$$ shows the amount of vitamins $$A,B$$, and $$C$$ (in milligrams per cup) for each type of food. Matrix $$N$$ shows the number of cups of each type of food consumed by the dog each day. $$M=\begin{array}{cc} & \begin{array}{ccc} \text{Kibble} & \text{Bits} & \text{Chunks}\\ \end{array}\\ \begin{array}{l} \text{Vitamin A}\\ \text{Vitamin B}\\ \text{Vitamin C} \end{array} & \left[\begin{array}{ccc} \hspace{1.2em}3\hspace{1.2em} & \hspace{1.2em}2\hspace{1.2em} & \hspace{1.2em}4\hspace{1.2em} \\ 2 & 4 & 5 \\ 2 & 5 & 1 \end{array}\right] \end{array}\quad$$$$N=\begin{array}{cc} & \begin{array}{c} \text{Cups}\\ \end{array}\\ \begin{array}{l} \text{Kibble} \\ \text{Bits} \\ \text{Chunks}\\ \end{array} & \left[\begin{array}{c} \hspace{.5em}1.2\hspace{.5em}\\ .8 \\5 \end{array}\right] \end{array}$$ Given $$A=MN$$, calculate the entry $$a_{21}$$ in matrix $$A$$ and give the correct interpretation for $$a_{21}$$. $$a_{21}=8.1\quad$$ Given this mixture of food, the dog will consume 8.1 mg of Vitamin B each day. If you would like to see more videos on this topic, click the following link and check the related videos. 18. The quantity demanded for a certain brand of portable CD players is 200 units when the unit price is set at \72. The quantity demanded increases by 1000 units when the unit price decreases by \40. The corresponding supply equation is $$p(x) = .06x+10$$ where $$p(x)$$ is the price in dollars at which $$x$$ CD players will be supplied. 1. Find the demand equation, assuming the demand equation is linear. 2. Find the equilibrium point and then interpret the meaning of the equilibrium point. 3. If the price is $$\60$$, will there be a surplus or shortage of CD players. Explain your reasoning.​ 4. How many CD players will be demanded if they are given away for free? 5. Suppliers will only provide the items if the price is above what value? 1. The demand equation is $$p_d=-0.04x+80.$$ 2. The equilibrium point is $$(700,52)$$. When the CD players are sold at 52 each, 700 CD players will be demanded and supplied. 3. There will be a surplus of CD players because the supply is greater than the demand when the price is 60. 4. 2000 CD players will be demanded if $$p=0$$ (the CD players are free). 5. The price must be above 10 for suppliers to supply any CD players. If you would like to see more videos on this topic, click the following link and check the related videos. 19. Given the matrix $$A$$ below, 1. pivot on the element in row 2, column 2. 2. use the calculator to put matrix $$A$$ in row reduced echelon form (RREF) and then find the solution, if one exists. Assume the variables are $$x$$ and $$y$$\[A=\left[\begin{array}{cc|c}1&-2&4\\[5pt] 0&7&-7\\[5pt] 2&-3&5 \end{array}\right]$ 1. $\left[\begin{array}{cc|c}1&0&2\\[5pt] 0&1&-1\\[5pt] 2&0&2 \end{array}\right]$ 2. $\left[\begin{array}{cc|c}1&0&0\\[5pt] 0&1&0\\[5pt] 0&0&1 \end{array}\right]$ There is no solution. If you would like to see more videos on this topic, click the following link and check the related videos. 20. When 150 items are produced and sold, a company has total costs of \$27,000 and the cost per item is $$85$$. The total profit from selling 150 units is $$\45,000$$.

1. Find the cost equation $$C(x).$$
2. Find the revenue equation $$R(x).$$
3. Find the profit equation $$P(x).$$

1. $$C(x)=85x+14250$$
2. $$R(x)=480x$$
3. $$P(x)=395x-14250$$

If you would like to see more videos on this topic, click the following link and check the related videos.

21. Write a system of equations that represents the following problem. Do not solve the system.
An executive of Trident Communications recently traveled to London, Paris, and Rome. He paid $215,$260, and $250 per night for lodging in London, Paris and Rome, respectively, and his hotel bills totaled$3420. He spend $100,$125, and $110 per day for his meals in London, Paris, and Rome, respectively, and his expenses for meals totaled$2000. If he spent twice as many days in London as he did in Paris and Rome combined, how many days did he stay in each city?

$$\begin{cases} 215x + 260y+250z=3420\$3pt] 100x+125y+110z=2000\\[3pt] x=2(y+z) \end{cases}$$ And $$x\geq0,$$ $$y\geq 0,$$ and $$z\geq 0.$$ If you would like to see more videos on this topic, click the following link and check the related videos. 22. Given a matrix $$A$$ below, use row operations to get the matrix in reduced row echelon form.\[\left[\begin{array}{cc|c}1&-2&4\\[5pt] 0&1&-1\\[5pt] 2&-3&5 \end{array}\right]$

$\left[\begin{array}{cc|c}1&0&2\\[5pt] 0&1&-1\\[5pt] 0&0&1 \end{array}\right]$