The system of equations below has solution . What is the value of ?

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 53 | Free SAT Prep | Aniko

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Question 53·Medium·Systems of Two Linear Equations in Two Variables

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The system of equations below has solution $(x, y)$ .

\begin{aligned} 4x + 3y &= 18 \\ 2x - y &= 4 \end{aligned}

What is the value of $x$ ?

For systems of two linear equations on the SAT, elimination is usually fastest: line up the equations, choose one variable to eliminate, and multiply one equation (if needed) so the coefficients of that variable are opposites. Add or subtract the equations to cancel that variable, solve the resulting one-variable equation, and then (if required) substitute back to find the other variable. Always do a quick mental check by plugging your solution into one of the original equations to catch arithmetic mistakes.

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Look at the structure of the system

You have two equations with the same two variables, $x$ and $y$ . Think about how to combine them so that one variable disappears.

Choose a variable to eliminate

Would it be easier to eliminate $x$ or $y$ ? Look at the coefficients: $4$ and $2$ for $x$ , and $3$ and $-1$ for $y$ .

Make coefficients match (or be opposites)

Try multiplying one of the equations so that the coefficients of $y$ become opposites. Then add the equations to cancel $y$ and solve for $x$ .

Check your solution

After you find $x$ , plug it back into one of the original equations to confirm it satisfies the system.

Desmos Guide

Rewrite each equation in slope-intercept form

Solve each equation for $y$ :

From $2x - y = 4$ , get $y = 2x - 4$ .
From $4x + 3y = 18$ , get $3y = 18 - 4x$ , so $y = -\tfrac{4}{3}x + 6$ .

Enter the equations into Desmos

Type y = 2x - 4 and y = (-4/3)x + 6 as two separate lines. Desmos will graph both lines on the same coordinate plane.

Find the intersection point

Tap or click the point where the two lines intersect. Look at the $x$ -coordinate of this intersection; that value is the solution for $x$ in the system.

Step-by-step Explanation

Write down the system and choose a method

We are given the system

\begin{aligned} 4x + 3y &= 18 \\ 2x - y &= 4 \end{aligned}

We want $x$ . A fast way is elimination: make the coefficients of $y$ opposites (or the same) so they cancel when you add the equations.

Create opposite coefficients for y

The second equation has $-y$ . If we multiply the entire second equation by $3$ , the coefficient of $y$ will become $-3y$ , which will cancel with the $+3y$ in the first equation.

Multiply the second equation by $3$ :

2x - y = 4

After multiplying by $3$ :

6x - 3y = 12

Now our system is

\begin{aligned} 4x + 3y &= 18 \\ 6x - 3y &= 12 \end{aligned}

Add the equations to eliminate y

Add the two new equations term by term so that $y$ cancels:

\begin{aligned} 4x + 3y &= 18 \\ 6x - 3y &= 12 \end{aligned}

Adding the equations gives:

10x = 30

Now you have a single equation with just $x$ .

Solve for x and match the choice

Solve $10x = 30$ by dividing both sides by $10$ :

\begin{aligned} 10x &= 30 \\ x &= 3 \end{aligned}

So the value of $x$ is $3$ , which corresponds to choice B) 3.

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