The system of equations is The solution to the system is . What is the value of ?

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SAT Systems of Two Linear Equations Question 112 | Free SAT Prep | Aniko

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Question 112·Easy·Systems of Two Linear Equations in Two Variables

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The system of equations is

\begin{cases} x + y = 12 \\ x = 3y \end{cases}

The solution to the system is $(x, y)$ . What is the value of $x$ ?

For systems of equations where one equation already has a variable isolated (like x = 3y), use substitution to be efficient: substitute that expression into the other equation to get a single-variable equation, solve it, then plug back to find the other variable. Finally, double-check which variable the question asks for (x or y) so you report the correct one; you can quickly confirm your result by graphing both equations and checking their intersection point.

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Decide which equation to start with

One of the equations already has x isolated. How can you use that to rewrite the other equation using only y?

Substitute for x

In the equation $x + y = 12$ , replace $x$ with the expression given in the other equation. What new equation in terms of y do you get?

Solve for y first

After substituting, you should have an equation like something times y equals 12. Solve that for y.

Answer the question being asked

Once you know y, plug it back into $x = 3y$ . Be careful: the question asks for the value of x, not y.

Desmos Guide

Enter the equations

In Desmos, type the first equation as x + y = 12 on one line and the second equation as x = 3y on another line. Both lines should appear on the graph.

Locate the intersection point

Adjust the viewing window if needed so you can see where the two lines cross. Tap or click on the intersection point; Desmos will display its coordinates $(x, y)$ .

Read off the x-value

Look at the x-coordinate of the intersection point. That x-value is the solution to the system and the answer choice you should select.

Step-by-step Explanation

Use substitution to write an equation in one variable

You are given the system

\begin{cases} x + y = 12 \\ x = 3y \end{cases}

Because the second equation says $x = 3y$ , substitute $3y$ for $x$ in the first equation:

3y + y = 12.

Solve for y

Combine like terms on the left side of

3y + y = 12

to get

4y = 12.

Now divide both sides by 4:

y = 3.

Use the relationship between x and y

You know from the system that $x = 3y$ and from the previous step that $y = 3$ .

Substitute $y = 3$ into $x = 3y$ :

x = 3y = 3 \cdot 3.

Find the value of x

Evaluate the product from the previous step:

x = 3 \cdot 3 = 9.

So the value of $x$ is $9$ , which corresponds to choice C.

Strategy

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation