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

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 119 | Free SAT Prep | Aniko

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Question 119·Medium·Systems of Two Linear Equations in Two Variables

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Solve the following system of equations.

\begin{aligned} 2x + 7y &= 37 \\ 3x - 7y &= -2 \end{aligned}

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

Your answer

(Express the answer as an integer)

For systems of two linear equations, quickly scan the coefficients to see if one variable can be eliminated by simply adding or subtracting the equations. If one variable has coefficients that are equal and opposite, add the equations to cancel that variable, giving a one-step equation in the other variable. Solve that simple equation, then (if needed) substitute back to find the other variable—but always double-check which variable the question actually asks for so you report the correct one.

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Look for an easy elimination

Compare the coefficients of $x$ and $y$ in the two equations. Which variable has coefficients that are the same number but with opposite signs?

Combine the equations

Try adding the two equations together. What happens to the $y$ terms when you do this?

Solve the resulting one-variable equation

After $y$ is eliminated, you should have an equation of the form $5x = 35$ . How do you solve this for $x$ ?

Desmos Guide

Enter the equations into Desmos

Type each equation exactly as given into Desmos on separate lines:

2x + 7y = 37
3x - 7y = -2 Desmos can graph these as implicit equations and will show where they intersect.

Find the intersection point

On the graph, locate the point where the two lines cross. Click or tap on that intersection point; Desmos will display its coordinates $(x,y)$ .

Read off the required value

From the displayed intersection coordinates, note the $x$ -coordinate of the point. That $x$ -value is the solution the question is asking for.

Step-by-step Explanation

Notice which variable is easy to eliminate

Look at the coefficients of $y$ in the two equations:

In the first equation, the coefficient of $y$ is $7$ .
In the second equation, the coefficient of $y$ is $-7$ .

Because these are the same number with opposite signs, adding the two equations will cancel $y$ immediately.

Add the two equations to eliminate $y$

Add the left sides and the right sides of the equations:

(2x + 7y) + (3x - 7y) = 37 + (-2)

Combine like terms on the left and right:

$2x + 3x = 5x$
$7y - 7y = 0$
$37 + (-2) = 35$

So you get a single equation in $x$ :

5x = 35

Solve for $x$

Now solve the simple equation $5x = 35$ by dividing both sides by $5$ :

x = \frac{35}{5}

So the value of $x$ is $7$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Step-by-step Explanation