Questions: Write the equation of a line perpendicular to 4x - 3y = -8 that passes through the point (-4,2). The equation of the line is y = (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.).

$Write the equation of a line perpendicular to 4x - 3y = -8 that passes through the point (-4,2). The equation of the line is y = (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.).$

Transcript text: Write the equation of a line perpendicular to $4 x-3 y=-8$ that passes through the point $(-4,2)$. The equation of the line is $y=$ $\square$ (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.).

Solution

Solution Steps

Step 1: Determine the Slope of the Original Line

The original line is given by the equation $4x - 3y = -8$. To find its slope, we first convert it to the slope-intercept form $y = mx + b$.

\[ 3y = 4x + 8 \implies y = \frac{4}{3}x + \frac{8}{3} \]

Thus, the slope $m$ of the original line is $\frac{4}{3}$.

Step 2: Find the Slope of the Perpendicular Line

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line is:

\[ m_{\text{perpendicular}} = -\frac{3}{4} \]

Step 3: Use the Point-Slope Form to Find the Equation

We use the point-slope form of the equation of a line, $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point $(-4, 2)$ through which the perpendicular line passes.

\[ y - 2 = -\frac{3}{4}(x + 4) \]

Step 4: Simplify to Slope-Intercept Form

Simplifying the equation:

\[ y - 2 = -\frac{3}{4}x - 3 \]

Adding 2 to both sides:

\[ y = -\frac{3}{4}x - 1 \]