Question

the equation of line j is 8x + 3y = -12. line k, which is perpendicular to line j, includes the point (-4, -1). what is the equation of line k?
write the equation in slope-intercept form. write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Explanation:

Step1: Find slope of line j

Rewrite \(8x + 3y=-12\) in slope - intercept form \(y = mx + b\) (where \(m\) is the slope).
Subtract \(8x\) from both sides: \(3y=-8x - 12\).
Divide by 3: \(y=-\frac{8}{3}x-4\). So slope of line \(j\), \(m_j = -\frac{8}{3}\).

Step2: Find slope of line k

If two lines are perpendicular, the product of their slopes is \(- 1\). Let slope of line \(k\) be \(m_k\).
Then \(m_j\times m_k=-1\). Substitute \(m_j = -\frac{8}{3}\): \(-\frac{8}{3}\times m_k=-1\).
Solve for \(m_k\): \(m_k=\frac{3}{8}\) (multiply both sides by \(-\frac{3}{8}\)).

Step3: Use point - slope form to find equation of line k

Point - slope form is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(-4,-1)\) and \(m = \frac{3}{8}\).
Substitute values: \(y-(-1)=\frac{3}{8}(x - (-4))\).
Simplify: \(y + 1=\frac{3}{8}(x + 4)\).

Step4: Convert to slope - intercept form

Distribute \(\frac{3}{8}\): \(y + 1=\frac{3}{8}x+\frac{3}{2}\).
Subtract 1 from both sides: \(y=\frac{3}{8}x+\frac{3}{2}-1\).
Simplify \(\frac{3}{2}-1=\frac{3 - 2}{2}=\frac{1}{2}\). So \(y=\frac{3}{8}x+\frac{1}{2}\).

Answer:

\(y=\frac{3}{8}x+\frac{1}{2}\)