Question

a line with a slope of $-\frac{5}{4}$ passes through the points $(4, -10)$ and $(k, 0)$. what is the value of $k$?
$k = \square$
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Explanation:

Step1: Recall the slope formula

The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \( m = -\frac{5}{4} \), \( (x_1,y_1)=(4,-10) \) and \( (x_2,y_2)=(k,0) \). So we substitute these values into the slope formula:
\( -\frac{5}{4}=\frac{0 - (-10)}{k - 4} \)

Step2: Simplify the numerator

Simplify the numerator of the right - hand side: \( 0-(-10)=0 + 10 = 10 \). So the equation becomes:
\( -\frac{5}{4}=\frac{10}{k - 4} \)

Step3: Cross - multiply

Cross - multiply to get rid of the fractions. We have \( - 5(k - 4)=4\times10 \)

Step4: Distribute and solve for k

First, distribute the - 5 on the left - hand side: \( -5k+20 = 40 \)
Subtract 20 from both sides: \( -5k=40 - 20 \)
\( -5k=20 \)
Divide both sides by - 5: \( k=\frac{20}{-5}=-4 \)

Answer:

\( k=-4 \)