The system of equations is What is the solution to the given system of equations?

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 76 | Free SAT Prep | Aniko

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Question 76·Easy·Systems of Two Linear Equations in Two Variables

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The system of equations is

$x + y = 10$

$x - y = 2$

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

For systems like this on the SAT, quickly scan for coefficients that are opposites (like $+y$ and $-y$ ). If you see them, use elimination: add or subtract the equations to cancel one variable, solve for the remaining variable, then substitute back to find the other. Finally, do a fast mental check that your ordered pair satisfies both equations to avoid trap answers that only work in one equation.

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Look at how $y$ appears in both equations

In one equation you have $+y$ and in the other you have $-y$ . What happens to $y$ if you add the two equations together?

Try combining the equations

Add the left-hand sides and the right-hand sides of the two equations. Which variable disappears, and what equation in one variable do you get?

Once you know one variable, find the other

After you solve for $x$ , plug that value into either $x + y = 10$ or $x - y = 2$ to solve for $y$ .

Check your pair

Whichever ordered pair you find, make sure it makes both equations true, not just one of them.

Desmos Guide

Enter the two equations as lines

Type y = 10 - x for the first equation and y = x - 2 for the second equation. Desmos will graph both lines.

Find the intersection point

Click or tap where the two lines cross. Desmos will display the coordinates of this intersection. Those coordinates give the values of $x$ and $y$ that solve the system.

Step-by-step Explanation

Notice the structure of the system

The system is:

x + y = 10 \\[0.7em] x - y = 2

The coefficients of $y$ are $+1$ and $-1$ , which makes this system perfect for the elimination method: adding the equations will cancel out $y$ .

Add the two equations to eliminate $y$

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

(x + y) + (x - y) = 10 + 2

On the left, $+y$ and $-y$ cancel:

2x = 12

Now solve for $x$ :

x = 6

Keep this value of $x$ for the next step.

Substitute $x$ back to find $y$

Use either original equation. Using $x + y = 10$ :

Substitute $x = 6$ :

6 + y = 10

Solve for $y$ :

y = 4

Now you know both $x$ and $y$ .

State and check the solution

Write the solution as an ordered pair $(x, y)$ using the values you found:

(6,4)

Check in both equations:

$x + y = 6 + 4 = 10$ ✔️
$x - y = 6 - 4 = 2$ ✔️

So the solution to the system is $(6,4)$ .

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