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.).

Solution Steps

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 \(\frac{4}{3}\).

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:

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

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.

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

Simplifying the equation:

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

Adding 2 to both sides:

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

Final Answer