A solution to the given system of equations is . What is the greatest possible value of ?

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SAT Nonlinear Equations & Systems Question 202 | Free SAT Prep | Aniko

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Question 202·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

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\begin{aligned} y &= x \\ (x - 1)^2 + (y - 8)^2 &= 37 \end{aligned}

A solution to the given system of equations is $(x, y)$ . What is the greatest possible value of $x$ ?

For systems where a line intersects a circle or other curve, use substitution to reduce the system to a single equation in one variable. Replace one variable using the simple equation (like $y=x$ ), simplify carefully to get a quadratic, and solve it by factoring or using the quadratic formula. When you get two solutions, always double-check what the question is asking (for example, greatest or least value, or a specific variable) so you choose the correct root quickly.

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Connect the two equations

The first equation says $y=x$ . How can you use that to rewrite the second equation so it only has $x$ in it?

After substitution, simplify carefully

Once you substitute $y=x$ into the second equation, you will have two squared binomials in $x$ . Expand both squares and combine like terms.

Recognize the type of equation

After simplifying, you should have a quadratic equation in $x$ . Think about how to solve it (factoring or quadratic formula), and remember that a quadratic usually has two solutions.

Answer exactly what is asked

You will get two values for $x$ . Read the question again to decide which of those two values to report.

Desmos Guide

Graph the line

In Desmos, enter the first equation as y = x to graph the line.

Graph the circle

On a new line in Desmos, enter the second equation as (x - 1)^2 + (y - 8)^2 = 37 to graph the circle.

Find the intersection points

Use Desmos to click on the points where the line and the circle intersect. Desmos will show you the coordinates of those intersection points; note the $x$ -values of both points.

Choose the correct x-value

Compare the two $x$ -values from the intersection points and identify the larger one. That greater $x$ -value is the answer to the question.

Step-by-step Explanation

Use the fact that y = x

From the first equation, $y=x$ . This means that wherever you see $y$ in the second equation, you can replace it with $x$ .

Substitute $y=x$ into the circle equation:

(x - 1)^2 + (y - 8)^2 = 37 \Rightarrow (x - 1)^2 + (x - 8)^2 = 37.

Expand and simplify to get a quadratic

Now expand each squared term:

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

(x - 8)^2 = x^2 - 16x + 64

Add them together and set equal to 37:

x^2 - 2x + 1 + x^2 - 16x + 64 = 37

Combine like terms:

2x^2 - 18x + 65 = 37

Subtract 37 from both sides:

2x^2 - 18x + 28 = 0

Divide everything by 2:

x^2 - 9x + 14 = 0.

Solve the quadratic equation for x

You now have the quadratic equation

x^2 - 9x + 14 = 0.

Factor this quadratic. You need two numbers that multiply to 14 and add to 9. Once factored, this equation will give you two possible values of $x$ .

Find the two x-values and choose the greater one

The factoring is

x^2 - 9x + 14 = (x - 7)(x - 2) = 0.

So the solutions are $x = 7$ and $x = 2$ .

The question asks for the greatest possible value of $x$ , so the correct choice is $x = 7$ .

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Step-by-step Explanation

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Step-by-step Explanation