Consider the system of equations: Which of the following ordered pairs satisfies the system?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 201 | Free SAT Prep | Aniko

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Question 201·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Consider the system of equations:

\begin{aligned} y &= x^2 - 4x \\ y &= 2x - 5 \end{aligned}

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

For systems with one linear and one quadratic equation, use substitution by setting the expressions equal (since both are $y =$ something) to reduce the problem to a single quadratic in $x$ . Solve the quadratic—usually by factoring on the SAT—and then substitute each $x$ back into one of the original equations to get the corresponding $y$ -values. Finally, match the resulting $(x,y)$ pairs to the answer choices, and if needed, quickly verify by plugging a choice back into both equations to avoid algebra mistakes.

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What does it mean to satisfy both equations?

A solution to the system must have an $x$ and $y$ that work in both equations at the same time, not just one of them. Keep that in mind as you test or solve.

Use the fact that both equations are solved for y

Since both equations are written as $y =$ (in terms of $x$ ), set the expressions $x^2 - 4x$ and $2x - 5$ equal to each other to get a single equation in $x$ .

Solve and then plug back in

After setting the expressions equal, you’ll get a quadratic equation. Factor it (if possible) to find the possible $x$ -values, then substitute each $x$ into one of the original equations to find the corresponding $y$ and compare with the answer choices.

Desmos Guide

Graph the parabola

In Desmos, enter the first equation exactly as y = x^2 - 4x to graph the parabola.

Graph the line

Enter the second equation as y = 2x - 5 to graph the line on the same coordinate plane.

Find the intersection points

Look for the points where the line and parabola cross. Click or tap each intersection point; Desmos will display its coordinates.

Match an intersection to an answer choice

Compare the coordinates of the intersection points shown by Desmos with the listed answer choices, and choose the option whose $(x, y)$ matches one of those intersection coordinates.

Step-by-step Explanation

Set the two expressions for y equal

Because both equations are written as $y =$ something, any solution must give the same $y$ -value in both equations. So set the right-hand sides equal to each other:

x^2 - 4x = 2x - 5.

Now solve this equation for $x$ .

Solve the quadratic equation for x

Move all terms to one side:

\begin{aligned} x^2 - 4x &= 2x - 5 \\ x^2 - 4x - 2x + 5 &= 0 \\ x^2 - 6x + 5 &= 0. \end{aligned}

Factor the quadratic:

x^2 - 6x + 5 = (x - 1)(x - 5) = 0,

so the possible $x$ -values are $x = 1$ or $x = 5$ .

Find the corresponding y-values and match to the choices

Use either original equation (for example, the linear one $y = 2x - 5$ ) to find $y$ for each $x$ :

If $x = 1$ :
- $y = 2(1) - 5 = -3$ .
If $x = 5$ :
- $y = 2(5) - 5 = 5$ .

So the system’s solutions are the points $(1,-3)$ and $(5,5)$ . Among the answer options, only $(1,-3)$ appears, so $(1,-3)$ is the ordered pair that satisfies the system.

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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