SAT Nonlinear Equations & Systems Question 118 | Free SAT Prep | Aniko

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Question 118·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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Which of the following ordered pairs (x, y) satisfies the system of equations?

\begin{aligned} y &= x^2 - 1 \\ x + y &= 11 \end{aligned}

For systems where one equation already has a variable isolated (like $y = x^2 - 1$ ), use substitution: plug that expression into the other equation to get a single equation in one variable, solve it (factoring or quadratic formula if needed), then back-substitute to find the other variable. On multiple-choice questions, you can quickly verify by plugging each candidate pair into both original equations and checking which one satisfies both.

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Start with the equation that is already solved for a variable

Notice that the first equation is $y = x^2 - 1$ . How can you use this expression for $y$ in the second equation $x + y = 11$ ?

Form a single equation in one variable

After you substitute $y = x^2 - 1$ into $x + y = 11$ , you should get an equation involving only $x$ . Rearrange it so that one side equals $0$ .

Solve and check

Solve the resulting quadratic equation for $x$ , then plug each $x$ back into $y = x^2 - 1$ to find $y$ . Finally, see which ordered pair from the answer choices matches one of the solutions you found.

Desmos Guide

Enter both equations in Desmos

Type y = x^2 - 1 as your first equation. For the second equation, first solve for $y$ to get $y = 11 - x$ , then type y = 11 - x as your second equation.

Find the intersection points

Look at the graph where the parabola and the line cross. Tap or click on each intersection point to see its coordinates; these are the solutions to the system.

Compare with the answer choices

Compare the intersection coordinates you see in Desmos with the listed answer options and identify which ordered pair from the graph appears among the choices.

Step-by-step Explanation

Use substitution to make one equation

You are given the system

\begin{aligned} y &= x^2 - 1 \\ x + y &= 11 \end{aligned}

The first equation already tells you $y$ in terms of $x$ . Substitute $y = x^2 - 1$ into the second equation:

x + (x^2 - 1) = 11

Now you have one equation with just $x$ .

Rearrange and solve the quadratic for x

Simplify and move all terms to one side:

\begin{aligned} x + x^2 - 1 &= 11 \\ x^2 + x - 1 - 11 &= 0 \\ x^2 + x - 12 &= 0 \end{aligned}

Factor the quadratic:

x^2 + x - 12 = (x + 4)(x - 3) = 0

So the possible $x$ -values are $x = -4$ and $x = 3$ .

Find the corresponding y-values and match to a choice

Use $y = x^2 - 1$ to find $y$ for each $x$ :

If $x = 3$ :
- $y = 3^2 - 1 = 9 - 1 = 8$ , so one solution pair is $(3, 8)$ .
If $x = -4$ :
- $y = (-4)^2 - 1 = 16 - 1 = 15$ , so the other solution pair is $(-4, 15)$ .

Now compare these solution pairs to the answer choices. Only $(3, 8)$ appears in the list, so the ordered pair that satisfies the system is $(3, 8)$ .

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