Lesson 9 Eating Up the Lunchroom Budget Develop Understanding

Ready

The pairs of equations given are equivalent. Determine what was done to the first equation to get the second equation. (For example, everything multiplied by $5$ or Multiplicative Property of Equality) If more than one operation was performed, please indicate the operations and the order they were performed.

1.

$\begin{matrix} x + y = 5 \\ 3 x + 3 y = 15 \end{matrix}$

2.

$\begin{matrix} 4 x + 3 y = 12 \\ x + \frac{3}{4} y = 3 \end{matrix}$

3.

$\begin{matrix} 6 x + 4 y = 20 \\ 4 y = - 6 x + 20 \end{matrix}$

Determine whether the pairs of equations given are equivalent. If equivalent, state the operations used to create the second from the first. If not equivalent, show why not.

4.

$\begin{matrix} 54 x - 42 y = 90 \\ 9 x + 7 y = 15 \end{matrix}$

5.

$\begin{matrix} 12 x + 9 y = 21 \\ 4 x + 3 y = 7 \end{matrix}$

6.

$\begin{matrix} 2 x + 5 y = 10 \\ y = \frac{2}{5} x + 10 \end{matrix}$

Literal Equations: Solve each of the equations below for $y$ . Show your work and justifications.

7.

$- 5 x + y = 12$	$Justification$

8.

$6 x + 2 y = 12$	$Justification$

9.

$- 12 x + 4 y = - 16$	$Justification$

10.

$5 x - 2 y = 10$	$Justification$

Set

The equipment manager for the school athletics department is attempting to restock some of the needed uniform and equipment items for the upcoming baseball and football seasons. Based on current levels of inventory and the number of players returning, more socks, pants, and helmets will be needed. The equipment manager has organized the information in the matrix shown.

$\begin{array}{cc} \begin{matrix} \end{matrix} & \begin{matrix} Socks & Pants & Helmets \end{matrix} \\ \begin{matrix} Baseball \\ Football \end{matrix} & [\begin{matrix} 13 & 15 & 7 \\ 24 & 45 & 20 \end{matrix}] \end{array}$

The school has contracted with two supply stores in the past for equipment needs. The matrix given shows how much each store charges for the needed items.

$\begin{array}{rc} \begin{array}{cc} Big Sky & Play It \\ Sporting Goods & Forever \end{array} \\ \begin{array}{rcc} Cost per pair of socks \\ Cost per pair of pants \\ Cost per helmet \end{array} & [\begin{array}{cccc} $ 3.50 & $ 3.00 \\ $ 35.00 & $ 40.00 \\ $ 22.00 & $ 45.50 \end{array}] \end{array}$

11.

Calculate the values of $a$ , $b$ , $c$ , and $d$ in the “Total Cost Matrix.”

Total Cost Matrix:

$\begin{array}{rc} \begin{array}{cc} Big Sky & Play It \\ Sporting Goods & Forever \end{array} \\ \begin{array}{rcc} Baseball \\ Football \end{array} & [\begin{array}{cccc} a & b \\ c & d \end{array}] \end{array}$

12.

Show the detailed calculations for finding the value of $a$ and $b$ . How do you use the numbers in the first two matrices above to obtain the values for the “Total Costs Matrix”?

Go

Determine whether the functions represented are linear, exponential, or neither. State the type of function, and explain why it is what you indicate or show why on the representation provided. Solve the problems.

13.

Use the values in the table to predict $g (7)$ .

$x$	$g (x)$
$- 1$	$11$
$0$	$10$
$1$	$11$
$2$	$14$
$3$	$19$

14.

Where is this function increasing?

15.

What $x$ -value produces an output of $6$ ?

16.

$f (n) = 3 (n - 5) + 7$

What does $n$ equal when $f (n) = 22$ ?

17.

What explicit equation can be used to represent this table of values?

$x$	$h (x)$
$- 3$	$7$
$0$	$- 2$
$4$	$- 14$
$9$	$- 29$

18.

$y = (5) 3^{x}$

If this function models the size in acres of a wildfire for each hour that passes, how many square feet will have burned within $4$ hours?