Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ At a candy store, Miranda bought $2$ kilograms of jelly beans and $1$ kilogram of gummy worms for $\$13$ . Meanwhile, Vicky bought $1$ kilogram of gummy worms for $\$5$ . How much does the candy cost? $\newline$ A kilogram of jelly beans costs $\$$ _, and a kilogram of gummy worms costs $\$$ _.

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Grade 8

Solve a system of equations using any method: word problems

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

\newline

At a candy store, Miranda bought

2

kilograms of jelly beans and

1

kilogram of gummy worms for

\$13

. Meanwhile, Vicky bought

1

kilogram of gummy worms for

\$5

. How much does the candy cost?

\newline

A kilogram of jelly beans costs

\$

_____, and a kilogram of gummy worms costs

\$

_____.

Define Variables: Let $x$ be the cost per kilogram of jelly beans and $y$ be the cost per kilogram of gummy worms. We can write two equations based on the information given: $\newline$ Equation $1$ : $2x + y = 13$ (Miranda's purchase) $\newline$ Equation $2$ : $y = 5$ (Vicky's purchase)
Create Augmented Matrix: To solve using an augmented matrix, we first write the coefficients of the variables and the constants in a matrix form: $\begin{bmatrix} 2 & 1 & \vert & 13 \ 0 & 1 & \vert & 5 \end{bmatrix}$
Perform Row Operations: Since the second equation already has $y$ isolated, we can use it to substitute into the first equation. But let's stick to the matrix method and perform row operations to solve for $x$ and $y$ .
Solve for $x$ : We can use the second row to make the $y$ coefficient in the first row zero. To do this, we multiply the second row by $-1$ and add it to the first row: $\newline$ (-1)\cdot\begin{bmatrix}0 & 1 | & 5\end{bmatrix} + \begin{bmatrix}2 & 1 | & 13\end{bmatrix} = \begin{bmatrix}2 & 0 | & 8\end{bmatrix}$\newlineNow the matrix looks like: $\newline$ \$\begin{bmatrix}2 & 0 | & 8\0 & 1 | & 5\end{bmatrix}$
Solve for y: From the first row of the new matrix, we have $2x = 8$. Dividing both sides by $2$ gives us $x = 4$.
Final Cost Calculation: From the second row of the matrix, we have $y = 5$, which we already knew from Vicky's purchase.
Final Cost Calculation: From the second row of the matrix, we have $y = 5$, which we already knew from Vicky's purchase.So, a kilogram of jelly beans costs $\$4$, and a kilogram of gummy worms costs $\$5$.