Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ Julian and Lee went to an arcade where the machines took tokens. Julian played $4$ games of skee ball and $3$ games of pinball, using a total of $17$ tokens. At the same time, Lee played $1$ game of pinball, using up $3$ tokens. How many tokens does each game require? $\newline$ Every game of skee ball requires $\_\_\_\_$ tokens, and every game of pinball requires $\_\_\_\_$ tokens.

Full solution

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

\newline

Julian and Lee went to an arcade where the machines took tokens. Julian played

4

games of skee ball and

3

games of pinball, using a total of

17

tokens. At the same time, Lee played

1

game of pinball, using up

3

tokens. How many tokens does each game require?

\newline

Every game of skee ball requires

\_\_\_\_

tokens, and every game of pinball requires

\_\_\_\_

tokens.

Define Variables: Let $x$ be the number of tokens required for a game of skee ball and $y$ for a game of pinball. Julian's games give us the equation $4x + 3y = 17$ .
Julian's Equation: Lee's single game of pinball gives us the equation $y = 3$ .
System of Equations: We now have the system of equations: $4x + 3y = 17$ $y = 3$
Augmented Matrix: To solve using an augmented matrix, we write the coefficients of the variables and the constants in a matrix: $\begin{bmatrix} 4 & 3 & | & 17 \ 0 & 1 & | & 3 \end{bmatrix}$
Eliminate Variable: Since the second equation is already solved for $y$ , we can use it to eliminate $y$ from the first equation. We'll multiply the second row by $-3$ and add it to the first row.
Multiply and Add: Multiplying the second row by $-3$ gives us: $\newline$ $[0 -3 | -9]$
Updated Matrix: Adding this to the first row: $\newline$ $[4 0 | 8]$
Solve for $x$ : Now we have the matrix: $\begin{bmatrix} 4 & 0 & \vert & 8 \ 0 & 1 & \vert & 3 \end{bmatrix}$
Solve for y: From the first row of the matrix, we have $4x = 8$ , which gives us $x = 2$ .
Final Token Requirements: We already know $y = 3$ from the second row of the matrix.
Final Token Requirements: We already know $y = 3$ from the second row of the matrix.So, every game of skee ball requires $2$ tokens, and every game of pinball requires $3$ tokens.