What is the solution to the following system of equations?

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SAT Systems of Two Linear Equations Question 43 | Free SAT Prep | Aniko

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Question 43·Easy·Systems of Two Linear Equations in Two Variables

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What is the solution $(x, y)$ to the following system of equations?

\begin{aligned} 3x + y &= 7 \\ x - y &= 1 \end{aligned}

For systems like this, scan the coefficients to see if one variable can be eliminated easily by adding or subtracting the equations; here, $y$ has coefficients $+1$ and $-1$ , so adding the equations cancels $y$ immediately and gives $x$ in one step. After finding one variable, substitute it into either original equation to find the other, and quickly check that your ordered pair satisfies both equations before selecting your answer.

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Look for easy elimination

Compare the coefficients of $y$ in the two equations. Is there a way to add or subtract the equations so that $y$ cancels out?

Try combining the equations

What happens if you add the two equations together, left side with left side and right side with right side? Which variable disappears?

Use your first result to get the second variable

Once you find a value for $x$ , plug it back into either original equation to solve for $y$ .

Desmos Guide

Enter the equations

In Desmos, type 3x + y = 7 on one line and x - y = 1 on another line so both lines are graphed.

Find the intersection point

Look for the point where the two lines cross. You can tap or click near the crossing point and Desmos will display its coordinates.

Match the intersection to the answer choices

The coordinates of that intersection are the $(x, y)$ solution to the system. Compare those coordinates to the answer options and choose the matching ordered pair.

Step-by-step Explanation

Decide how to combine the equations

Look at the system:

$3x + y = 7$
$x - y = 1$

Notice that the coefficients of $y$ are $+1$ and $-1$ . If you add the two equations, the $y$ terms will cancel out, which makes it easy to solve for $x$ .

Add the equations to eliminate y and solve for x

Add the left sides and the right sides:

Left side: $(3x + y) + (x - y) = 3x + x + y - y = 4x$
Right side: $7 + 1 = 8$

So you get the equation $4x = 8$ . Divide both sides by 4:

$x = 2$ .

Substitute to find y and write the solution

Now plug $x = 2$ into either original equation. Use $x - y = 1$ :

$2 - y = 1$
Subtract 2 from both sides: $-y = -1$
Multiply both sides by $-1$ : $y = 1$

So the solution that satisfies both equations is $(x, y) = (2, 1)$ .

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