Consider the system of equations: Which ordered pair satisfies the system?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 229 | Free SAT Prep | Aniko

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Question 229·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Consider the system of equations:

y=x^2-6x+8

y=x-2

Which ordered pair $(x,y)$ satisfies the system?

For systems where one equation is linear and the other is quadratic, equate the two expressions (since they both equal $y$ ), rearrange to form a quadratic equal to 0, and solve it by factoring or using the quadratic formula. Then plug each resulting $x$ -value into the simpler equation (usually the line) to find the corresponding $y$ , and finally choose the ordered pair from the answer choices that matches one of these intersection points and satisfies both original equations.

Hints

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Use the idea that both equations equal y

Since both equations are written as $y = \text{(expression)}$ , think about how you can combine them to get an equation in just $x$ .

Form a single equation in x

Set $x^2 - 6x + 8$ equal to $x - 2$ and rearrange to get a quadratic equation equal to 0.

Solve and then plug back in

After you solve the quadratic for $x$ , use the linear equation $y = x - 2$ to find $y$ for each $x$ -value, then see which resulting point appears in the choices.

Desmos Guide

Graph both equations

In the first expression line, type y = x^2 - 6x + 8. In the second expression line, type y = x - 2. You should see a parabola and a straight line on the same coordinate plane.

Find the intersection points

Click (or tap) where the line and parabola intersect. Desmos will show the coordinates of each intersection point; note both of these coordinate pairs.

Match with the answer choices

Compare the intersection coordinates you found in Desmos with the listed options and choose the option that exactly matches one of those intersection points.

Step-by-step Explanation

Interpret what it means to solve the system

A solution to the system is an ordered pair $(x,y)$ that makes both equations true at the same time. Geometrically, it is a point where the parabola $y=x^2-6x+8$ and the line $y=x-2$ intersect.

Set the equations equal to each other

Both expressions equal $y$ , so we can set them equal:

x^2 - 6x + 8 = x - 2

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

x^2 - 6x + 8 - x + 2 = 0 \\[0.7em] x^2 - 7x + 10 = 0

Solve the quadratic for x

Factor the quadratic $x^2 - 7x + 10$ :

x^2 - 7x + 10 = (x - 5)(x - 2) = 0

Set each factor equal to 0:

$x - 5 = 0$ gives $x = 5$ .
$x - 2 = 0$ gives $x = 2$ .

So there are two possible $x$ -values for intersection: $x=2$ and $x=5$ .

Find the corresponding y-values and match to the choices

Use the simpler equation $y = x - 2$ to find $y$ for each $x$ :

For $x = 2$ : $y = 2 - 2 = 0$ , so one intersection point is $(2, 0)$ .
For $x = 5$ : $y = 5 - 2 = 3$ , so another intersection point is $(5, 3)$ .

Only one of these points appears among the answer choices, so the ordered pair that satisfies the system is $(5, 3)$ .

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