SAT Linear Equations in Two Variables Question 75 | Free SAT Prep | Aniko

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Question 75·Hard·Linear Equations in Two Variables

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A bookstore sold a total of 90 books in one day. Each fiction book sold for $12, and each nonfiction book sold for $18. Let $x$ be the number of fiction books sold and let $y$ be the number of nonfiction books sold. The total number of books and the total sales in dollars can be modeled by the system of equations

\begin{aligned} x + y &= 90 \\ 12x + 18y &= 1500. \end{aligned}

According to this model, how many nonfiction books were sold?

Your answer

(Express the answer as an integer)

For SAT system-of-equations word problems, first write clear equations for totals (like total items and total cost) and label each variable with its meaning. Then choose the fastest method—usually substitution if one equation is already solved for a variable, or elimination if the coefficients line up easily. After solving, always match the variable back to what the question asks for (fiction vs. nonfiction, adults vs. children, etc.) and quickly plug your values into both original equations to ensure they satisfy the totals.

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Use the total number of books

Start with $x + y = 90$ . Can you rewrite this to express $x$ in terms of $y$ , or $y$ in terms of $x$ ?

Substitute into the money equation

Once you have $x$ written in terms of $y$ , replace $x$ in the equation $12x + 18y = 1500$ with that expression and simplify.

Solve the resulting one-variable equation

After substitution, you should have an equation with only $y$ . Combine like terms, isolate $y$ , and solve carefully to avoid arithmetic mistakes.

Answer exactly what is asked

The variable $y$ represents the number of nonfiction books. Once you solve for $y$ , that value directly answers the question.

Desmos Guide

Enter the system of equations

In Desmos, type x + y = 90 on one line and 12x + 18y = 1500 on another line so that both lines (equations) are graphed.

Locate the intersection point

Look for the point where the two lines intersect and tap/click it; Desmos will display the coordinates $(x, y)$ of this intersection. The $y$ -coordinate of this point is the number of nonfiction books sold.

Step-by-step Explanation

Understand what each equation represents

The equation $x + y = 90$ says the total number of books sold is 90, where $x$ is fiction books and $y$ is nonfiction books. The equation $12x + 18y = 1500$ says that fiction books at $12 each and nonfiction books at $18 each together brought in $1500.

Express one variable in terms of the other

From $x + y = 90$ , solve for $x$ in terms of $y$ :

x + y = 90 \\[0.7em] x = 90 - y

Now you can replace $x$ in the second equation with $90 - y$ .

Substitute and solve for y

Substitute $x = 90 - y$ into $12x + 18y = 1500$ :

12(90 - y) + 18y = 1500

Distribute 12:

1080 - 12y + 18y = 1500

Combine like terms for $y$ :

1080 + 6y = 1500

Subtract 1080 from both sides:

6y = 1500 - 1080 = 420

Now divide both sides by 6 to get $y$ .

Interpret the solution and answer the question

Divide to solve for $y$ :

y = \frac{420}{6} = 70

Remember that $y$ is the number of nonfiction books sold, so according to the model, 70 nonfiction books were sold.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation