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 .

Explanation:

Step1: Find slope of given line

Rewrite \(4x - 3y=-3\) in slope - intercept form \(y = mx + b\) (where \(m\) is the slope).
Solve for \(y\):
\(-3y=-4x - 3\)
\(y=\frac{4}{3}x + 1\)
The slope of the given line \(m_1=\frac{4}{3}\).

Step2: Find slope of perpendicular line

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

Step3: Use point - slope form to find equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(-1,-6)\) and \(m =-\frac{3}{4}\).
Substitute the values:
\(y-(-6)=-\frac{3}{4}(x - (-1))\)
\(y + 6=-\frac{3}{4}(x + 1)\)

Step4: Convert to slope - intercept form

Expand the right - hand side:
\(y+6=-\frac{3}{4}x-\frac{3}{4}\)
Subtract 6 from both sides:
\(y=-\frac{3}{4}x-\frac{3}{4}-6\)
\(y=-\frac{3}{4}x-\frac{3}{4}-\frac{24}{4}\)
\(y=-\frac{3}{4}x-\frac{27}{4}\)

Answer:

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