Question

question 7 of 40
write an equation of the line satisfying the given conditions. write the answer in slope - intercept form or standard form. express numbers as integers or simplified fractions.
the line contains the point (-1, -6) and is perpendicular to 4x - 3y = -3.
the equation of the line is
question 8 of 40

Explanation:

Step1: Find slope of given line

Rewrite \(4x - 3y=-3\) in slope - intercept form (\(y = mx + b\)).
Subtract \(4x\) from both sides: \(-3y=-4x - 3\).
Divide by \(-3\): \(y=\frac{4}{3}x + 1\). The slope (\(m_1\)) of this line is \(\frac{4}{3}\).

Step2: Find slope of perpendicular line

If two lines are perpendicular, the product of their slopes is \(- 1\) (\(m_1\times m_2=-1\)).
Let \(m_2\) be the slope of the required line. Then \(\frac{4}{3}\times m_2=-1\), so \(m_2=-\frac{3}{4}\).

Step3: Use point - slope form to find equation

Point - slope form is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(-1,-6)\) and \(m =-\frac{3}{4}\).
Substitute values: \(y-(-6)=-\frac{3}{4}(x - (-1))\), which simplifies to \(y + 6=-\frac{3}{4}(x + 1)\).

Step4: Convert to slope - intercept form

Distribute \(-\frac{3}{4}\): \(y+6=-\frac{3}{4}x-\frac{3}{4}\).
Subtract 6 from both sides: \(y=-\frac{3}{4}x-\frac{3}{4}-6\).
Convert 6 to fourths: \(6=\frac{24}{4}\), so \(y=-\frac{3}{4}x-\frac{3}{4}-\frac{24}{4}=-\frac{3}{4}x-\frac{27}{4}\).

Answer:

\(y =-\frac{3}{4}x-\frac{27}{4}\)