What is the solution to the system of equations?

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 13 | Free SAT Prep | Aniko

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Question 13·Easy·Systems of Two Linear Equations in Two Variables

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$2x - y = 7$

$x + y = 11$

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

For systems of two linear equations on the SAT, first check if elimination is easy: if one variable has opposite coefficients (like $y$ and $-y$ ), add or subtract the equations to eliminate that variable quickly, then solve for the remaining one and substitute back. Always verify that your solution satisfies both equations. When answer choices are given as ordered pairs, you can also plug each pair into both equations and eliminate wrong options fast, but algebraic elimination is usually faster and more reliable under time pressure.

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Choose a method

Look at the two equations and decide whether substitution or elimination will be quicker. Do you see any variables that can cancel if you add or subtract the equations?

Try eliminating a variable

Compare the $y$ terms in both equations: one is $-y$ and the other is $+y$ . What happens to $y$ if you add the two equations together?

Solve step by step

After you eliminate one variable and solve for the other, plug that value back into either original equation to find the remaining variable.

Check your result

Once you find a pair $(x,y)$ , substitute it into both original equations to make sure it works in each one.

Desmos Guide

Enter the first equation in Desmos

Rewrite the first equation in slope-intercept form as $y = 2x - 7$ . In Desmos, type y = 2x - 7 to graph this line.

Enter the second equation in Desmos

Rewrite the second equation as $y = 11 - x$ . In Desmos, type y = 11 - x to graph the second line.

Locate the intersection point

Look at the graph where the two lines cross. Click on the intersection point; Desmos will display its coordinates. Those coordinates give the $(x,y)$ pair that solves the system.

Step-by-step Explanation

Write the system clearly

We are given the system

\begin{aligned} 2x - y &= 7 \\ x + y &= 11 \end{aligned}

Our goal is to find the pair $(x,y)$ that makes both equations true at the same time.

Eliminate one variable by adding the equations

Notice that in the first equation we have $-y$ and in the second equation we have $+y$ . If we add the two equations, the $y$ terms will cancel out:

\begin{aligned} (2x - y) + (x + y) &= 7 + 11 \\ 2x - y + x + y &= 18 \\ 3x &= 18 \end{aligned}

Now we can solve this simpler equation for $x$ .

Solve for $x$

From $3x = 18$ , divide both sides by $3$ :

x = \frac{18}{3} = 6

Now that we know $x$ , we will substitute this value into one of the original equations to find $y$ .

Substitute $x$ to find $y$ and state the solution

Use the second equation $x + y = 11$ and substitute $x = 6$ :

6 + y = 11

Subtract $6$ from both sides:

y = 11 - 6 = 5

So the solution that makes both equations true is $(x,y) = (6,5)$ , which corresponds to answer choice A.

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