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Hanson is at a ball game with his family. He goes to the concession stand to buy soft pretzels and water bottles for everyone. A water bottle costs $2\$$ , and a pretzel costs $3.50\$$ . Hanson buys $16$ items and spends $47\$$ in all. $\newline$ This system of equations can be used to represent the situation: $\newline$ $x + y = 16$ $\newline$ $2x + 3.5y = 47$ $\newline$ Which statement is correct? $\newline$ (A) In the system of equations, $x$ represents the amount of money Hanson spends on pretzels, and $y$ represents the amount of money he spends on water bottles. $\newline$ (B) In the system of equations, $x$ represents the number of water bottles Hanson buys, and $y$ represents the number of pretzels he buys. $\newline$ (C) In the system of equations, $x$ represents the number of pretzels Hanson buys, and $y$ represents the number of water bottles he buys. $\newline$ (D) In the system of equations, $x$ represents the amount of money Hanson spends on water bottles, and $y$ represents the amount of money he spends on pretzels. $\newline$ How much money does Hanson spend on pretzels? $\newline$ $3.50\$$ $2$ $3.50\$$ $3$

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Math Problems

Grade 7

Evaluate two-variable equations: word problems

Full solution

Q. Hanson is at a ball game with his family. He goes to the concession stand to buy soft pretzels and water bottles for everyone. A water bottle costs

2\$

, and a pretzel costs

3.50\$

. Hanson buys

16

items and spends

47\$

in all.

\newline

This system of equations can be used to represent the situation:

\newline

x + y = 16

\newline

2x + 3.5y = 47

\newline

Which statement is correct?

\newline

(A) In the system of equations,

x

represents the amount of money Hanson spends on pretzels, and

y

represents the amount of money he spends on water bottles.

\newline

(B) In the system of equations,

x

represents the number of water bottles Hanson buys, and

y

represents the number of pretzels he buys.

\newline

x

represents the number of pretzels Hanson buys, and

y

represents the number of water bottles he buys.

\newline

(D) In the system of equations,

x

represents the amount of money Hanson spends on water bottles, and

y

represents the amount of money he spends on pretzels.

\newline

How much money does Hanson spend on pretzels?

\newline

3.50\$

2

3.50\$

3

Identify Variables: First, identify what $x$ and $y$ represent in the system of equations. From the choices given, $(B)$ is correct: $x$ represents the number of water bottles Hanson buys, and $y$ represents the number of pretzels Hanson buys.
Set Up Equations: Set up the equations based on the problem statement: $\newline$ $1$ . $x + y = 16$ (Total items bought) $\newline$ $2$ . $2x + 3.5y = 47$ (Total cost of items)
Solve for x: Solve the first equation for $x$ : $x = 16 - y$
Substitute $x$ : Substitute $x$ in the second equation: $\newline$ $2(16 - y) + 3.5y = 47$ $\newline$ $32 - 2y + 3.5y = 47$
Combine Like Terms: Combine like terms: $\newline$ $1.5y + 32 = 47$
Solve for y: Solve for y: $\newline$ $1.5y = 47 - 32$ $\newline$ $1.5y = 15$
Find $y$ Value: Divide both sides by $1.5$ to find $y$ : $y = \frac{15}{1.5}$ $y = 10$
Substitute $y$ : Substitute $y$ back into the equation for $x$ : $x = 16 - 10$ $x = 6$
Calculate Total Cost: Calculate the total cost spent on pretzels: $\newline$ Cost of pretzels = $3.5 \times y$ $\newline$ Cost of pretzels = $3.5 \times 10$ $\newline$ Cost of pretzels = $35$ (\$)