Question

what is the equation of the line in slope - intercept form that is perpendicular to the line $y = \frac{3}{4}x - 2$ and passes through the point $(-12, 10)$?\
$y = \frac{4}{3}x + 10$\
$y = \frac{4}{3}x + 26$\
$y = -\frac{4}{3}x + 6$\
$y = -\frac{4}{3}x - 6$

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = \frac{3}{4}x - 2 \) is \( m_1=\frac{3}{4} \). For a line perpendicular to this, the slope \( m_2 \) is the negative reciprocal, so \( m_2 = -\frac{4}{3} \).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(-12,10) \) and \( m = -\frac{4}{3} \).
Substitute the values: \( y - 10=-\frac{4}{3}(x + 12) \)

Step3: Simplify to slope - intercept form

Expand the right - hand side: \( y - 10=-\frac{4}{3}x-16 \)
Add 10 to both sides: \( y=-\frac{4}{3}x - 16 + 10 \)
Simplify: \( y=-\frac{4}{3}x-6 \)
Wait, there is a mistake above. Let's re - calculate step 2 and 3.

Step1: Correct slope of perpendicular line

The slope of the given line \( y=\frac{3}{4}x - 2 \) is \( m=\frac{3}{4} \). The slope of a line perpendicular to it is the negative reciprocal. So \( m_{\perp}=-\frac{4}{3} \) (this step is correct).

Step2: Use point - slope form

The point - slope formula is \( y - y_1=m(x - x_1) \), with \( (x_1,y_1)=(-12,10) \) and \( m = -\frac{4}{3} \)
\( y-10 = -\frac{4}{3}(x+12) \)
Expand the right side: \( y - 10=-\frac{4}{3}x-16 \)
Add 10 to both sides: \( y=-\frac{4}{3}x-16 + 10\)
\( y=-\frac{4}{3}x - 6 \)

Wait, but let's check the options again. Wait, maybe I made a mistake in the slope. Wait, no, the negative reciprocal of \( \frac{3}{4} \) is \( -\frac{4}{3} \). Let's check the options. The option with slope \( -\frac{4}{3} \) and let's verify with the point.

Let's take the option \( y = -\frac{4}{3}x-6 \). When \( x=-12 \), \( y=-\frac{4}{3}\times(-12)-6=16 - 6 = 10 \), which matches the point \( (-12,10) \).

Wait, but let's re - do the calculation of the point - slope form correctly.

\( y - y_1=m(x - x_1) \), \( x_1=-12,y_1 = 10,m=-\frac{4}{3} \)

\( y-10=-\frac{4}{3}(x-(-12))=-\frac{4}{3}(x + 12) \)

\( y-10=-\frac{4}{3}x-16 \)

\( y=-\frac{4}{3}x-16 + 10=-\frac{4}{3}x-6 \)

Yes, this is correct.

Answer:

The equation of the line is \( y = -\frac{4}{3}x-6 \) (the option with this equation, looking at the options, it's the left - most option: \( y = -\frac{4}{3}x-6 \))