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Question 123·Hard·Linear Equations in Two Variables

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In the $xy$ -plane, a line $\ell$ passes through the point $(4,\,k)$ and is perpendicular to the line $3x - 2y = 7$ . The $x$ -intercept of $\ell$ is $(k+1,\,0)$ . What is the value of $k$ ?

Your answer

For perpendicular-line questions, first rewrite any standard-form equation into slope-intercept form to read off the slope quickly. Take the negative reciprocal to get the perpendicular slope. Then, if a second line is defined by points with variables, use the slope formula in terms of those variables and set it equal to the known perpendicular slope. Solve the resulting linear equation carefully, watching negative signs, and, if time allows, plug your value back into the coordinates to verify that the computed slope is indeed perpendicular.

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Identify the slope of the given line

Rewrite $3x - 2y = 7$ in the form $y = mx + b$ to find its slope $m$ .

Use the perpendicular slope rule

For two perpendicular lines, how is one slope related to the other? Once you know the slope of $3x - 2y = 7$ , determine the slope of line $\ell$ .

Write the slope of ℓ using its two points

Use the slope formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ with the points $(4, k)$ and $(k+1, 0)$ to express the slope of $\ell$ in terms of $k$ .

Create and solve an equation for k

Set your slope expression from the two points equal to the perpendicular slope you found, then solve the resulting equation for $k$ .

Desmos Guide

Represent the slope of ℓ as a function of k

In Desmos, let $x$ play the role of $k$ . Type the function f(x) = -x/(x - 3); this represents the slope of the line through $(4, x)$ and $(x+1, 0)$ .

Graph the perpendicular slope as a constant function

On a new line, type g(x) = -2/3 to graph a horizontal line showing the required perpendicular slope.

Find the k-value from the intersection

Look for the intersection point of the graphs of $f(x)$ and $g(x)$ . The $x$ -coordinate of this intersection gives the value of $k$ that makes the slopes match; read that value from the graph.

Step-by-step Explanation

Find the slope of the given line

Rewrite the line $3x - 2y = 7$ in slope-intercept form:

\begin{aligned} 3x - 2y &= 7 \\ -2y &= -3x + 7 \\ y &= \frac{3}{2}x - \frac{7}{2} \end{aligned}

The slope of this line is $\frac{3}{2}$ .

Use the perpendicular slope relationship

If two non-vertical lines are perpendicular, their slopes are negative reciprocals.

The given line has slope $\frac{3}{2}$ , so the slope of line $\ell$ must be

-\frac{1}{(3/2)} = -\frac{2}{3}.

So the slope of $\ell$ is $-\frac{2}{3}$ .

Express the slope of line ℓ using its two points

Line $\ell$ passes through $(4, k)$ and has $x$ -intercept $(k+1, 0)$ .

The slope of $\ell$ using these two points is

\text{slope} = \frac{0 - k}{(k+1) - 4} = \frac{-k}{k - 3}.

Set the slopes equal and form an equation

The slope we just found must equal the perpendicular slope $-\frac{2}{3}$ .

So set

\frac{-k}{k - 3} = -\frac{2}{3}.

Now solve this equation for $k$ by cross-multiplying.

Solve for k and conclude

Solve

\frac{-k}{k - 3} = -\frac{2}{3}

by cross-multiplying:

\begin{aligned} -3k &= -2(k - 3) \\ -3k &= -2k + 6 \\ -3k + 2k &= 6 \\ -k &= 6 \\ k &= -6 \end{aligned}

Therefore, the value of $k$ is $-6$ .

Question 123·Hard·Linear Equations in Two Variables

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