SAT Nonlinear Equations & Systems Question 130 | Free SAT Prep | Aniko

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Question 130·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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\begin{cases} x^{2}-y = 2 \\ y = -x \end{cases}

The ordered pair $(x, y)$ satisfies the system of equations above. Which of the following is a possible value of $x$ ?

For systems where one equation is already solved for a variable (like $y=-x$ ), use substitution: plug that expression into the other equation to get a single equation in one variable. Simplify carefully, solve the resulting quadratic by factoring if possible (it’s usually faster than the quadratic formula on the SAT), and then match your solutions to the answer choices—often you only need to identify which solution appears, not compute the full ordered pair.

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Choose which equation to substitute

One of the equations already solves for a variable: $y = -x$ . Think about how you can use this to eliminate $y$ from the other equation.

Substitute into the first equation

Replace $y$ in the equation $x^2 - y = 2$ with $-x$ . Carefully simplify the resulting expression.

Solve the quadratic and compare with choices

After substitution, you should get a quadratic equation in $x$ . Factor it (or use another method) to find all possible $x$ -values, then see which of those appears in the answer choices.

Desmos Guide

Enter the two equations

Rewrite the first equation in terms of $y$ : from $x^2 - y = 2$ get $y = x^2 - 2$ . In Desmos, enter y = x^2 - 2 and y = -x as two separate graphs.

Find the intersection points

Use Desmos to find the intersection points of the two graphs (tap/click where they cross or use the intersection feature). Note the $x$ -coordinates of these intersection points.

Compare with the choices

Look at the $x$ -values you found in Desmos and compare them to the answer options $-3$ , $-2$ , $0$ , and $2$ to determine which option is possible.

Step-by-step Explanation

Use substitution to get one equation in one variable

Start with the system:

\begin{cases} x^2 - y = 2 \\ y = -x \end{cases}

The second equation tells you $y=-x$ . Substitute $-x$ for $y$ in the first equation:

x^2 - (-x) = 2.

Simplify and form a quadratic equation

Simplify the left side:

x^2 - (-x) = x^2 + x.

So the equation becomes:

x^2 + x = 2.

Bring all terms to one side:

x^2 + x - 2 = 0.

Factor the quadratic

Factor $x^2 + x - 2$ :

x^2 + x - 2 = (x + 2)(x - 1).

Set each factor equal to zero:

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

So the possible $x$ -values from the system are $x = -2$ and $x = 1$ .

Match your solutions with the answer choices

The solutions for $x$ are $-2$ and $1$ . The answer choices are $-3$ , $-2$ , $0$ , and $2$ . The only value that matches one of our solutions is $-2$ , so the correct choice is $-2$ .

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