The system of equations is When the equations above are graphed in the -plane, what are the coordinates of the points of intersection of the two graphs?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 98 | Free SAT Prep | Aniko

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Question 98·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The system of equations is

y = x^2 + 4x + 1

y = x + 5

When the equations above are graphed in the $xy$ -plane, what are the coordinates $(x, y)$ of the points of intersection of the two graphs?

For systems where you have a line and a parabola, equate the two expressions for $y$ to get a single equation in $x$ , then rearrange to standard quadratic form and factor (or use the quadratic formula) to find the $x$ -values. Use the simpler equation—usually the line—to find each corresponding $y$ , and finally match the ordered pairs to the choices. If you’re unsure, you can quickly verify by substituting each choice’s points into both original equations to see which ones satisfy both.

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Relate intersection to the equations

At a point of intersection, the same $(x,y)$ pair lies on both graphs. What does that tell you about the two expressions that are both equal to $y$ ?

Set up and simplify the equation

Set $x^2 + 4x + 1$ equal to $x + 5$ and move everything to one side so you have a standard quadratic equation equal to $0$ . Can you factor it?

Use the simpler equation for y

Once you know the $x$ -values that work, plug each into the linear equation $y = x + 5$ to find the corresponding $y$ -values. Then form the ordered pairs and compare with the choices.

Desmos Guide

Graph the parabola

In Desmos, enter the first equation exactly as given: y = x^2 + 4x + 1. This will draw the parabola.

Graph the line

Enter the second equation: y = x + 5. This will draw the straight line on the same axes.

Find the intersection points

On the graph, click or tap where the line and parabola cross. Desmos will display the coordinates of each intersection point; read these coordinates carefully and match these coordinates to the answer choices.

Step-by-step Explanation

Use the idea of intersection

When two graphs intersect, they share the same point $(x,y)$ , which means they have the same $x$ -value and the same $y$ -value at that point.

Here, both expressions equal $y$ :

$y = x^2 + 4x + 1$
$y = x + 5$

So at any intersection, the right-hand sides must be equal:

x^2 + 4x + 1 = x + 5

Form and solve the quadratic equation for x

Move all terms to one side to get a quadratic equation:

\begin{aligned} x^2 + 4x + 1 &= x + 5 \\ x^2 + 4x + 1 - x - 5 &= 0 \\ x^2 + 3x - 4 &= 0 \end{aligned}

Now factor the quadratic:

x^2 + 3x - 4 = (x + 4)(x - 1) = 0

So the possible $x$ -values at intersection are:

\begin{aligned} x + 4 &= 0 \Rightarrow x = -4 \\ x - 1 &= 0 \Rightarrow x = 1 \end{aligned}

Find the corresponding y-values using the line

To find the $y$ -values for each $x$ , substitute into the simpler equation $y = x + 5$ .

For $x = 1$ :

y = 1 + 5

For $x = -4$ :

y = -4 + 5

These give the $y$ -values that go with each $x$ at the intersection points.

Finish the arithmetic and match to an answer choice

Complete the calculations from the previous step:

For $x = 1$ : $y = 1 + 5 = 6$ , so one intersection point is $(1,6)$ .
For $x = -4$ : $y = -4 + 5 = 1$ , so the other intersection point is $(-4,1)$ .

Therefore, the graphs intersect at $(1,6)$ and $(-4,1)$ , which corresponds to choice D.

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