Consider the system of equations Which set of ordered pairs represents all solutions to this system?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 91 | Free SAT Prep | Aniko

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Question 91·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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Consider the system of equations

\begin{cases} y = (x-2)^2 + 1 \\ y = 2x - 3 \end{cases}

Which set of ordered pairs represents all solutions $(x,y)$ to this system?

For a system where both equations are solved for y, especially when one is linear and the other is a parabola, quickly set the right-hand sides equal to form a single equation in x. Simplify to a quadratic, solve it (usually by factoring if possible), then substitute each x-value back into the easier equation to get the corresponding y-values. Finally, choose the answer that lists all resulting ordered pairs, remembering that a line and a parabola can intersect at up to two points.

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Connect the two equations

Both equations equal y. How can you use that fact to write a single equation involving only x?

Look for a quadratic

After you set the right-hand sides equal, expand $(x-2)^2$ and move all terms to one side. What kind of equation do you get?

Solve for x, then for y

Once you have the quadratic equation, solve it (by factoring or another method). Then, for each x-value you find, plug it back into one of the original equations to get the corresponding y-value.

Remember there may be more than one solution

A line and a parabola can intersect in 0, 1, or 2 points. Make sure you find all intersection points, not just the first one you see.

Desmos Guide

Graph the parabola

In the first expression line, type y = (x-2)^2 + 1 to graph the parabola.

Graph the line

In the second expression line, type y = 2x - 3 to graph the line on the same axes.

Find the intersection points

Zoom or pan as needed so both graphs are clearly visible where they cross. Click or tap each intersection point; Desmos will display the coordinates of these points.

Match with the answer choices

The intersection coordinates are the solutions to the system. Compare the set of all intersection points you see in Desmos to the answer choices and select the choice whose set of ordered pairs matches them.

Step-by-step Explanation

Set the equations equal

Both equations equal y:

First: $y = (x-2)^2 + 1$
Second: $y = 2x - 3$

Since they are both equal to y, set the right-hand sides equal to each other:

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

Simplify to a quadratic equation

Expand $(x-2)^2$ and simplify:

(x-2)^2 = x^2 - 4x + 4

So the equation becomes:

x^2 - 4x + 4 + 1 = 2x - 3

Combine like terms on the left:

x^2 - 4x + 5 = 2x - 3

Move everything to one side to get a quadratic equal to 0:

x^2 - 4x + 5 - 2x + 3 = 0

So:

x^2 - 6x + 8 = 0

Solve the quadratic for x

Factor the quadratic:

x^2 - 6x + 8 = (x - 2)(x - 4) = 0

Set each factor equal to 0:

$x - 2 = 0$ gives $x = 2$ .
$x - 4 = 0$ gives $x = 4$ .

So there are two x-values that satisfy the system: $x = 2$ and $x = 4$ . These will lead to two solution points.

Find the corresponding y-values and list all solutions

Use either original equation (the linear one is usually easier) to find y for each x.

Using $y = 2x - 3$ :

If $x = 2$ :
- $y = 2(2) - 3 = 4 - 3 = 1$ , so one solution is $(2,1)$ .
If $x = 4$ :
- $y = 2(4) - 3 = 8 - 3 = 5$ , so another solution is $(4,5)$ .

The system has exactly these two solutions, so the correct set of ordered pairs is $\{(2,1),(4,5)\}$ , which corresponds to answer choice A.

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