Question

eddie asked his brother gerald to help him with his homework. eddies first problem requires him to write the equation of a line perpendicular to $13x - 17y = 7$ and passes through the point $(16, 8)$. eddies work is shown below. what will be geralds response to eddies work?

$13x - 17y = 7$
$y = \frac{13}{17}x - \frac{7}{17}$	step 2
$y - 8 = \frac{13}{17}(x - 16)$	step 3
$y = \frac{13}{17}x - \frac{72}{17}$	step 4

• \im sorry, eddie, but your answer is incorrect because you wrote the original problem down wrong.\
• \correct! great job!\
• \im sorry, eddie, but your answer is incorrect because your slope for your new equation in step 3 was supposed to be $-\frac{17}{13}$ and not $\frac{13}{17}$.\
• \im sorry, eddie, but your answer is incorrect because you switched the 8 and 16 in the point - slope formula for step 3.\

Explanation:

Step1: Find slope of original line

The original line is \(13x - 17y = 7\). Rewrite in slope - intercept form \(y=mx + b\) (where \(m\) is the slope).
Starting with \(13x-17y = 7\), we get \(- 17y=-13x + 7\), then \(y=\frac{13}{17}x-\frac{7}{17}\). So the slope of the original line \(m_1=\frac{13}{17}\).

Step2: Determine slope of perpendicular line

If two lines are perpendicular, the product of their slopes \(m_1\times m_2=- 1\). Let the slope of the perpendicular line be \(m_2\).
We have \(\frac{13}{17}\times m_2=-1\), so \(m_2 =-\frac{17}{13}\).

Step3: Analyze Eddie's work

Eddie used the slope \(\frac{13}{17}\) for the perpendicular line in Step 3 (the point - slope form \(y - y_1=m(x - x_1)\) where \((x_1,y_1)=(16,8)\)). But the correct slope for the perpendicular line should be \(-\frac{17}{13}\), not \(\frac{13}{17}\).

Answer:

"I'm sorry, Eddie, but your answer is incorrect because your slope for your new equation in Step 3 was supposed to be \(-\frac{17}{13}\) and not \(\frac{13}{17}\)."