Consider the system of equations Which of the following lists all ordered pairs that satisfy the system?

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SAT Nonlinear Equations & Systems Question 116 | Free SAT Prep | Aniko

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Question 116·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Consider the system of equations

\begin{cases} x^2 - y = 5\\ x + y = 7 \end{cases}

Which of the following lists all ordered pairs $(x, y)$ that satisfy the system?

For systems with one linear equation and one quadratic (or other nonlinear) equation, use substitution: solve the linear equation for one variable, substitute into the nonlinear equation, and solve the resulting single-variable equation completely (finding all roots). Then plug each root back into the linear equation to get the corresponding second coordinate, and quickly check both original equations before matching the full set of solution pairs to the answer choices.

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Start with the easier equation

One equation is linear ( $x + y = 7$ ) and one is not. Solve the linear equation for $y$ in terms of $x$ first.

Substitute into the other equation

After you write $y$ in terms of $x$ , plug that expression into $x^2 - y = 5$ so that the equation has only $x$ .

Expect a quadratic

When you substitute, you will get a quadratic equation in $x$ . Solve it carefully—there may be more than one value of $x$ .

Don’t stop after the first solution

For each value of $x$ you find, compute the corresponding $y$ using $x + y = 7$ , then check which answer choice lists all such ordered pairs.

Desmos Guide

Graph the first equation

Rewrite $x^2 - y = 5$ as $y = x^2 - 5$ and type y = x^2 - 5 into Desmos.

Graph the second equation

Rewrite $x + y = 7$ as y = 7 - x and enter y = 7 - x into Desmos.

Find the intersection points

Look for the points where the line and the parabola intersect. Use Desmos’s intersection tool (tap/click on the intersection points) and read off the coordinates; these ordered pairs are the solutions to the system.

Step-by-step Explanation

Use the linear equation to express one variable

From the second equation,

x + y = 7

solve for $y$ :

y = 7 - x.

This expression can now be substituted into the first equation.

Substitute into the first equation to get one variable

Substitute $y = 7 - x$ into the first equation $x^2 - y = 5$ :

x^2 - (7 - x) = 5.

Simplify inside the parentheses and combine like terms:

x^2 - 7 + x = 5, \\[0.7em] x^2 + x - 7 = 5.

Now subtract 5 from both sides:

x^2 + x - 12 = 0.

This is a quadratic equation in $x$ .

Solve the quadratic for x

Factor the quadratic:

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

Set each factor equal to 0:

$x + 4 = 0$ gives $x = -4$ .
$x - 3 = 0$ gives $x = 3$ .

So there are two possible $x$ -values.

Find the corresponding y-values and form ordered pairs

Use $y = 7 - x$ to find each $y$ :

If $x = 3$ , then $y = 7 - 3 = 4$ , giving the pair $(3, 4)$ .
If $x = -4$ , then $y = 7 - (-4) = 11$ , giving the pair $(-4, 11)$ .

Both pairs satisfy both original equations, so the complete solution set is $(3, 4)$ and $(-4, 11)$ .

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