A system of equations is given by One solution to the system satisfies . What is the value of ?

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SAT Nonlinear Equations & Systems Question 122 | Free SAT Prep | Aniko

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Question 122·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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A system of equations is given by

y = 3x - 2 \qquad\text{and}\qquad y = (x - 4)^2.

One solution to the system satisfies $x < 5$ . What is the value of $x$ ?

Your answer

(Express the answer as an integer)

For a system with a line and a parabola (or any two equations that both equal the same variable), set the right-hand sides equal to create one equation in a single variable. Simplify it into standard quadratic form $ax^2 + bx + c = 0$ , then solve efficiently—usually by factoring if the numbers are friendly, or by using the quadratic formula if not. Always check for any extra conditions in the question (like $x<5$ or a specific interval) and use them to select the appropriate solution from the possible roots.

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Use the fact that both equations equal y

Since both expressions equal $y$ , what equation can you write by setting the right-hand sides equal to each other?

Handle the squared term carefully

Expand $(x - 4)^2$ into a standard quadratic expression in $x$ before combining like terms.

Turn it into a standard quadratic and solve

After you expand, move everything to one side so it looks like $ax^2 + bx + c = 0$ . Then solve that quadratic (factoring may be easiest) and finally use the condition $x < 5$ to decide which solution to keep.

Desmos Guide

Enter the equations

In Desmos, type y = 3x - 2 on one line and y = (x - 4)^2 on another so that both graphs (a line and a parabola) appear.

Locate the intersection points

Adjust the zoom so you can see where the line and the parabola cross. There should be two intersection points.

Read the correct x-value from the graph

Click (or tap) on each intersection point so Desmos shows its coordinates. Identify the intersection whose x-coordinate is less than 5; that x-value is the solution you need.

Step-by-step Explanation

Set the equations equal to each other

Both equations equal $y$ :

y = 3x - 2

and

y = (x - 4)^2.

So at intersection points, the right sides must be equal:

3x - 2 = (x - 4)^2.

Expand and rearrange into a quadratic equation

Expand $(x - 4)^2$ :

(x - 4)^2 = x^2 - 8x + 16.

Substitute this into the equation:

3x - 2 = x^2 - 8x + 16.

Move all terms to one side to get $0$ on the other side:

0 = x^2 - 8x + 16 - 3x + 2.

Combine like terms:

x^2 - 11x + 18 = 0.

Factor the quadratic

Factor the quadratic $x^2 - 11x + 18$ .

We need two numbers that multiply to $18$ and add to $11$ . Those numbers are $2$ and $9$ .

So the factorization is:

x^2 - 11x + 18 = (x - 2)(x - 9) = 0.

Solve for x and apply the condition x < 5

Set each factor equal to $0$ :

From $x - 2 = 0$ , we get $x = 2$ .
From $x - 9 = 0$ , we get $x = 9$ .

The problem says the solution must satisfy $x < 5$ , so only $x = 2$ works.

Therefore, the value of $x$ is 2.

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Step-by-step Explanation

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