The system of equations is given by The two graphs intersect at two points. What is the -coordinate of the point with the greater -value?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 101 | Free SAT Prep | Aniko

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Question 101·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The system of equations is given by

\begin{cases} y = x^2 - 4x + 3 \\ y = 2x - 1 \end{cases}

The two graphs intersect at two points. What is the $x$ -coordinate of the point with the greater $x$ -value?

For systems involving a line and a parabola, set the two expressions for $y$ equal to get a single equation in $x$ , rearrange into standard quadratic form, and use the quadratic formula (or factoring if easy) to find both solutions. Then carefully interpret the question: if it asks for the larger or smaller solution, compare the two roots (for expressions like $a \pm b$ , the one with the plus sign is larger) and match the correct one to the answer choices.

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Use the fact that intersection points share the same y-value

At an intersection of two graphs, the coordinates satisfy both equations. How can you use this to connect $x^2 - 4x + 3$ and $2x - 1$ ?

Turn it into a single equation in x

Once the right sides are set equal, rearrange everything to one side so that the equation has the form $ax^2 + bx + c = 0$ .

Apply the quadratic formula and compare the two solutions

Use the quadratic formula to solve your quadratic. You will get two $x$ -values; think about which one is larger to answer the question.

Desmos Guide

Graph both equations

In Desmos, enter y = x^2 - 4x + 3 on one line and y = 2x - 1 on another line to display the parabola and the line.

Locate the intersection points

Click or tap where the graphs cross; Desmos will show the coordinates of each intersection point. Note the two $x$ -values and identify which one is larger.

Match the larger x-value to a choice

Take the larger $x$ -value you see in Desmos, write it in exact form (with a square root), and then select the answer choice that matches that expression.

Step-by-step Explanation

Set the equations equal at the intersection

At a point where the two graphs intersect, the $y$ -values are the same.

So set the right-hand sides equal:

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

Rearrange into standard quadratic form

Move all terms to one side to get a standard quadratic equation $ax^2 + bx + c = 0$ .

x^2 - 4x + 3 = 2x - 1 \\[0.7em] x^2 - 4x + 3 - 2x + 1 = 0 \\[0.7em] x^2 - 6x + 4 = 0

Solve the quadratic equation

Use the quadratic formula for $x^2 - 6x + 4 = 0$ , where $a=1$ , $b=-6$ , and $c=4$ .

The quadratic formula is

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substitute $a=1$ , $b=-6$ , and $c=4$ :

x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(4)}}{2(1)} \\[0.7em] x = \frac{6 \pm \sqrt{36 - 16}}{2} \\[0.7em] x = \frac{6 \pm \sqrt{20}}{2}

Simplify $\sqrt{20} = 2\sqrt{5}$ :

x = \frac{6 \pm 2\sqrt{5}}{2} = 3 \pm \sqrt{5}

So the two intersection $x$ -values are $3 + \sqrt{5}$ and $3 - \sqrt{5}$ .

Choose the greater x-value and match the choice

Between $3 + \sqrt{5}$ and $3 - \sqrt{5}$ , the greater value is $3 + \sqrt{5}$ because adding $\sqrt{5}$ is larger than subtracting it.

Therefore, the $x$ -coordinate of the point with the greater $x$ -value is $3 + \sqrt{5}$ , which corresponds to choice B.

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