A wildlife researcher models the height (in meters) of a tracking balloon above the ground seconds after launch with the equation . A second device reports the height using the equation . Which choice gives the ordered pair that satisfies both equations?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 58 | Free SAT Prep | Aniko

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Question 58·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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A wildlife researcher models the height $y$ (in meters) of a tracking balloon above the ground $x$ seconds after launch with the equation $y=x^2$ . A second device reports the height using the equation $y=4x-4$ .

Which choice gives the ordered pair $(x,y)$ that satisfies both equations?

When a system is given as $y=$ (expression) and $y=$ (expression), set the expressions equal to eliminate $y$ , solve the resulting quadratic for $x$ , and then substitute back into either equation to get $y$ . For multiple choice, quickly check each ordered pair by plugging its $x$ into both formulas and confirming the same $y$ results.

Hints

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Use the fact that both equal $y$

If $y=x^2$ and also $y=4x-4$ , then $x^2=4x-4$ .

Rewrite as a quadratic equation

Move all terms to one side so the equation equals $0$ .

Substitute back to get $y$

After you find $x$ , plug it into either equation to find the matching $y$ -value.

Desmos Guide

Enter both equations

In Desmos, enter $y=x^2$ and $y=4x-4$ .

Find the intersection

Click the intersection point of the two graphs (or use the intersection tool) to display its coordinates.

Read the coordinates

The displayed intersection coordinate is the solution $(x,y)$ to the system.

Step-by-step Explanation

Set the equations equal and solve for $x$

Because both equations equal $y$ , set them equal to each other:

\begin{aligned} x^2 &= 4x-4 \\ x^2-4x+4&=0 \\ (x-2)^2&=0 \end{aligned}

So $x=2$ .

Find $y$ and report the ordered pair

Substitute $x=2$ into either equation:

y=x^2=2^2=4

Therefore, the solution is $(2,4)$ .

Strategy

Hints

Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation