Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ A caterer is making cookie trays for upcoming holiday parties. This morning, she made $4$ small trays and $1$ large tray, which contain a total of $103$ cookies. In the afternoon, she made $4$ small trays and $2$ large trays, which contain a total of $158$ cookies. How many cookies do the different sized trays contain? $\newline$ The small tray contains $\_$ cookies and the large one contains $\_$ cookies.

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Grade 8

Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

A caterer is making cookie trays for upcoming holiday parties. This morning, she made

4

small trays and

1

large tray, which contain a total of

103

cookies. In the afternoon, she made

4

small trays and

2

large trays, which contain a total of

158

cookies. How many cookies do the different sized trays contain?

\newline

The small tray contains

\_

cookies and the large one contains

\_

cookies.

Equations Setup: Let's denote the number of cookies in a small tray as $S$ and the number of cookies in a large tray as $L$ . We can then write two equations based on the given information. $\newline$ Equation $1$ (from the morning batch): $4S + 1L = 103$ $\newline$ Equation $2$ (from the afternoon batch): $4S + 2L = 158$
Elimination Method: To use elimination, we want to eliminate one of the variables. We can do this by multiplying the first equation by $2$ , so that the coefficients of $L$ in both equations match. $\newline$ Multiplying the first equation by $2$ gives us: $8S + 2L = 206$
Subtracting Equations: Now we have the system of equations: $\newline$ $8S + 2L = 206$ $\newline$ $4S + 2L = 158$ $\newline$ We can subtract the second equation from the first to eliminate $L$ . $\newline$ $(8S + 2L) - (4S + 2L) = 206 - 158$
Solving for S: Performing the subtraction, we get: $\newline$ $8S - 4S + 2L - 2L = 206 - 158$ $\newline$ $4S = 48$
Substitute and Solve for L: Now we can solve for $S$ by dividing both sides of the equation by $4$ . $\frac{4S}{4} = \frac{48}{4}$ $S = 12$
Substitute and Solve for L: Now we can solve for $S$ by dividing both sides of the equation by $4$ .
$\frac{4S}{4} = \frac{48}{4}$
$S = 12$ Now that we have the value for $S$ , we can substitute it back into one of the original equations to solve for $L$ . We'll use the first equation:
$4S + 1L = 103$
$4(12) + L = 103$
Substitute and Solve for L: Now we can solve for $S$ by dividing both sides of the equation by $4$ .
$\frac{4S}{4} = \frac{48}{4}$
$S = 12$ Now that we have the value for $S$ , we can substitute it back into one of the original equations to solve for $L$ . We'll use the first equation:
$4S + 1L = 103$
$4(12) + L = 103$ Substitute $S = 12$ into the equation and solve for $L$ :
$4$ $0$
$4$ $1$
$4$ $2$