SAT Linear Equations in Two Variables Question 5 | Free SAT Prep | Aniko

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Question 5·Medium·Linear Equations in Two Variables

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A sporting-goods store sells baseball caps and knit hats. On a certain day, the store sold a total of 18 hats for a combined revenue of $258. The situation can be modeled by the system of equations

\begin{aligned} x + y &= 18 \\ 15x + 12y &= 258 \end{aligned}

where $x$ is the number of baseball caps sold and $y$ is the number of knit hats sold.

Which statement is the best interpretation of $12$ in this context?

For questions about interpreting numbers in a word problem model, first underline what each variable stands for in the story. Then locate the given number in the equation and interpret the whole term it appears in (for example, see that $12y$ represents something like "something per knit hat times number of knit hats"). From that term, isolate what the standalone number must mean, and pick the choice that matches that real-world meaning, being careful not to confuse counts (like number of hats) with rates (like dollars per hat) or totals (like total revenue).

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Focus on the right equation

Look at the second equation, $15x + 12y = 258$ , rather than the first one. Think about what the $258$ represents in the story.

Break the equation into parts

If $258$ is the total money from selling hats, what do the two parts $15x$ and $12y$ represent individually?

Zoom in on the term with $y$

Since $y$ is the number of knit hats sold, what real-world quantity does $12y$ represent? Then, what must the $12$ itself stand for in that product?

Desmos Guide

Enter the system of equations

Type the two equations into Desmos on separate lines:

x + y = 18
15x + 12y = 258 This will graph both lines and show their intersection point, which represents the actual numbers of each type of hat sold.

Look at how revenue is built from each term

In Desmos, note that the second equation uses the expression 15x + 12y to equal the total revenue 258. Think about how you normally compute revenue from sales: it is the product of an amount per item and the number of items, for each type, then added together.

Connect the coefficient 12 to the context

Focus on the part of the expression that involves knit hats, 12y. Since $y$ is the number of knit hats, consider what quantity must be multiplied by that number to give the money contributed by knit hats. That quantity is what the 12 represents in this context.

Step-by-step Explanation

Match variables to the story

The problem tells you what each variable means:

$x$ is the number of baseball caps sold.
$y$ is the number of knit hats sold.

So any term with $x$ involves baseball caps, and any term with $y$ involves knit hats.

Interpret the revenue equation

The second equation is

15x + 12y = 258.

The right side, $258$ , is the total money the store made from selling hats that day. That means the left side, $15x + 12y$ , must also represent the total revenue:

$15x$ is the money from selling $x$ baseball caps.
$12y$ is the money from selling $y$ knit hats.

Each term is "money from one type of hat" added together.

Interpret the number 12 specifically

Focus on the term $12y$ :

$y$ is the number of knit hats sold.
$12y$ is the total money from knit hats.

To get money from knit hats, you multiply

\text{(money per knit hat)} \times \text{(number of knit hats)} = \text{total money from knit hats}.

So the $12$ in $12y$ must be the money (in dollars) for each knit hat. Therefore, the best interpretation is: Each knit hat costs $12.

Strategy

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation