Question

consider the line $y = -3x + 4$. find the equation of the line that is parallel to this line and passes through the point $(2, 5)$. find the equation of the line that is perpendicular to this line and passes through the point $(2, 5)$. note that the aleks graphing calculator may be helpful in checking your answer. equation of parallel line: equation of perpendicular line:

Explanation:

Step1: Identify parallel line slope

Parallel lines have equal slopes. For $y=-3x+4$, slope $m_1=-3$.

Step2: Use point-slope for parallel line

Point-slope form: $y-y_1=m(x-x_1)$. Substitute $(x_1,y_1)=(2,5)$, $m=-3$:
$y-5=-3(x-2)$
Simplify to slope-intercept:
$y-5=-3x+6$
$y=-3x+11$

Step3: Identify perpendicular line slope

Perpendicular slopes are negative reciprocals: $m_2=\frac{1}{3}$.

Step4: Use point-slope for perpendicular line

Substitute $(x_1,y_1)=(2,5)$, $m=\frac{1}{3}$ into point-slope form:
$y-5=\frac{1}{3}(x-2)$
Simplify to slope-intercept:
$y-5=\frac{1}{3}x-\frac{2}{3}$
$y=\frac{1}{3}x+\frac{13}{3}$

Answer:

Equation of parallel line: $y=-3x+11$
Equation of perpendicular line: $y=\frac{1}{3}x+\frac{13}{3}$