Question

Consider the line y = - - X+4.4.***4.Find the equation of the line that is perpendicular to this line and passes through the point (-9, 6).Find the equation of the line that is parallel to this line and passes through the point (-9, 6).Note that the ALEKS graphing calculator may be helpful in checking your answer.Equation of perpendicular line: 0-Equation of parallel line:X0P

Answer 1

ANSWER:

$\begin{gathered} \text{perpendicular} \\ y=\frac{4}{3}x+18 \\ \text{parallel} \\ y=-\frac{3}{4}x-\frac{3}{4} \end{gathered}$

STEP-BY-STEP EXPLANATION:

We have that the equation of a line in its slope and y-intercept form is the following:

$\begin{gathered} y=mx+b \\ \text{where m is the slope and b is y-intercept} \end{gathered}$

Now, when two equations are perpendicular, the slope of both are opposite, that is, the product between them is equal to -1, like this:

$\begin{gathered} m_1\cdot m_2=-1 \\ -\frac{3}{4}\cdot m_2=-1 \\ m_2=\frac{4}{3} \end{gathered}$

We replace the values of the point (-9, 6) and the slope to calculate the value of the y -intercept:

$\begin{gathered} 6=-9\cdot\frac{4}{3}+b \\ b=12+6 \\ b=18 \end{gathered}$

Therefore, the perpendicular equation that also passes through the point (-9,6) is:

$y=\frac{4}{3}x+18$

Now, when two lines are parallel, the slope is the same, therefore we calculate directly that it passes through the point (-9, 6)

$\begin{gathered} 6=-9\cdot-\frac{3}{4}+b_{} \\ b=6+9\cdot-\frac{3}{4} \\ b=-\frac{3}{4} \end{gathered}$

therefore, the equation of the line that is parallel to this line and passes through the point (-9, 6) is:

$y=-\frac{3}{4}x-\frac{3}{4}$