11.8 Solving systems with cramer's rule  (Page 7/11)

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Men aged 20–29, 30–39, and 40–49 made up 78% of the population at a prison last year. This year, the same age groups made up 82.08% of the population. The 20–29 age group increased by 20%, the 30–39 age group increased by 2%, and the 40–49 age group decreased to $\text{\hspace{0.17em}}\frac{3}{4}\text{\hspace{0.17em}}$ of their previous population. Originally, the 30–39 age group had 2% more prisoners than the 20–29 age group. Determine the prison population percentage for each age group last year.

At a women’s prison down the road, the total number of inmates aged 20–49 totaled 5,525. This year, the 20–29 age group increased by 10%, the 30–39 age group decreased by 20%, and the 40–49 age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the 30–39 age group than the 20–29 age group. Determine the prison population for each age group last year.

20–29: 2,100, 30–39: 2,600, 40–49: 825

For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown in [link] .

Fat (g) Protein (g) Carbohydrates (g)
Almonds (10) 6 2 3
Cranberries (10) 0.02 0 8
Cashews (10) 7 3.5 5.5

For the special “low-carb”trail mix, there are 1,000 pieces of mix. The total number of carbohydrates is 425 g, and the total amount of fat is 570.2 g. If there are 200 more pieces of cashews than cranberries, how many of each item is in the trail mix?

For the “hiking” mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. If there is the same amount of almonds as cashews, how many of each item is in the trail mix?

300 almonds, 400 cranberries, 300 cashews

For the “energy-booster” mix, there are 1,000 pieces in the mix, containing 145 g of protein and 625 g of carbohydrates. If the number of almonds and cashews summed together is equivalent to the amount of cranberries, how many of each item is in the trail mix?

Systems of Linear Equations: Two Variables

For the following exercises, determine whether the ordered pair is a solution to the system of equations.

$\begin{array}{l}3x-y=4\\ x+4y=-3\text{\hspace{0.17em}}\end{array}$ and $\text{\hspace{0.17em}}\left(-1,1\right)$

No

$\begin{array}{l}6x-2y=24\\ -3x+3y=18\text{\hspace{0.17em}}\end{array}$ and $\text{\hspace{0.17em}}\left(9,15\right)$

For the following exercises, use substitution to solve the system of equations.

$\begin{array}{l}10x+5y=-5\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3x-2y=-12\hfill \end{array}$

$\left(-2,3\right)$

$\begin{array}{l}\frac{4}{7}x+\frac{1}{5}y=\frac{43}{70}\\ \frac{5}{6}x-\frac{1}{3}y=-\frac{2}{3}\end{array}$

$\begin{array}{l}5x+6y=14\\ 4x+8y=8\end{array}$

$\left(4,-1\right)$

For the following exercises, use addition to solve the system of equations.

$\begin{array}{l}3x+2y=-7\\ 2x+4y=6\end{array}$

$\begin{array}{r}3x+4y=2\\ 9x+12y=3\end{array}$

No solutions exist.

$\begin{array}{l}8x+4y=2\\ 6x-5y=0.7\end{array}$

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

A factory has a cost of production $\text{\hspace{0.17em}}C\left(x\right)=150x+15\text{,}000\text{\hspace{0.17em}}$ and a revenue function $\text{\hspace{0.17em}}R\left(x\right)=200x.\text{\hspace{0.17em}}$ What is the break-even point?

$\left(300,60,000\right)$

A performer charges $\text{\hspace{0.17em}}C\left(x\right)=50x+10\text{,}000,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the total number of attendees at a show. The venue charges \$75 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?

$\left(400,30,000\right)$

Systems of Linear Equations: Three Variables

For the following exercises, solve the system of three equations using substitution or addition.

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