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Maria and Nate graphed lines on a coordinate plane. Maria's line is represented by the equation y = -5/6x +8. Nate's line is perpendicular to Maria's line. Which of the following could be an equation for Nate's line?

a 5x-6y=15
b 5x+6y=15
c 6x-5y=15
d 6x+5y=15

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c) 6x - 5y = 15

Step-by-step explanation:

Slope-intercept form of a linear equation: $$y=mx+b$$

(where m is the slope and b is the y-intercept)

Maria's line: $$y=-\dfrac{5}{6}x+8$$

Therefore, the slope of Maria's line is $$-\frac{5}{6}$$

If two lines are perpendicular to each other, the product of their slopes will be -1.

Therefore, the slope of Nate's line (m) is:

\begin{aligned}\implies m \times -\dfrac{5}{6} &=-1\\m & =\dfrac{6}{5}\end{aligned}

Therefore, the linear equation of Nate's line is:

$$y=\dfrac{6}{5}x+b\quad\textsf{(where b is some constant)}$$

Rearranging this to standard form:

$$\implies y=\dfrac{6}{5}x+b$$

$$\implies 5y=6x+5b$$

$$\implies 6x-5y=-5b$$

Therefore, option c could be an equation for Nate's line.
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