Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Two friends visited a taffy shop. Tanvi bought $5$ kilograms of strawberry taffy and $5$ kilograms of banana taffy for $\$65$ . Next, Albert bought $5$ kilograms of strawberry taffy and $1$ kilogram of banana taffy for $\$41$ . How much does the candy cost? $\newline$ A kilogram of strawberry taffy costs $\$$ _, and a kilogram of banana taffy costs $\$$ _.

Full solution

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

\newline

Two friends visited a taffy shop. Tanvi bought

5

kilograms of strawberry taffy and

5

kilograms of banana taffy for

\$65

. Next, Albert bought

5

kilograms of strawberry taffy and

1

kilogram of banana taffy for

\$41

. How much does the candy cost?

\newline

A kilogram of strawberry taffy costs

\$

_____, and a kilogram of banana taffy costs

\$

_____.

Equations Setup: Let's denote the cost of a kilogram of strawberry taffy as $S$ dollars and the cost of a kilogram of banana taffy as $B$ dollars. We can write two equations based on the information given: $\newline$ For Tanvi: $5S + 5B = 65$ (Equation $1$ ) $\newline$ For Albert: $5S + 1B = 41$ (Equation $2$ ) $\newline$ We will use these equations to solve for $S$ and $B$ using the elimination method.
Elimination Method: To eliminate one of the variables, we can multiply Equation $2$ by $-5$ and add it to Equation $1$ to eliminate $S$ . $\newline$ Multiplying Equation $2$ by $-5$ gives us: $-25S - 5B = -205$ (Equation $3$ )
Combine Equations: Now we add Equation $1$ and Equation $3$ : $\newline$ $(5S + 5B) + (-25S - 5B) = 65 + (-205)$ $\newline$ This simplifies to: $-20S = -140$
Solve for S: We divide both sides of the equation by $-20$ to solve for S: $\newline$ $-20S / -20 = -140 / -20$ $\newline$ $S = 7$ $\newline$ So, a kilogram of strawberry taffy costs $\$7$ .
Substitute $S$ into Equation: Now that we have the value for $S$ , we can substitute it back into Equation $2$ to solve for $B$ : $5(7) + 1B = 41$ $35 + B = 41$
Final Solution: Subtracting $35$ from both sides gives us: $\newline$ $B = 41 - 35$ $\newline$ $B = 6$ $\newline$ So, a kilogram of banana taffy costs $\$6$ .