Question

1. write the equation in slope - intercept form of the line parallel to the equation y = -4x + 5 passing through (1, -6). work y = -4x - 2 29) write the equation in slope - intercept form of the line perpendicular to y = -2x + 5 and passes through (-4, 10). work y = \frac{1}{2}x + 12

Explanation:

Problem 28:

Step1: Recall slope - intercept form and parallel lines property

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. If two lines are parallel, they have the same slope. The given line is $y=-4x + 5$, so the slope of the parallel line $m=-4$.

Step2: Use the point - slope form to find $b$

We know the line passes through the point $(1,-6)$ and has a slope $m = - 4$. The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(1,-6)$ and $m=-4$.
Substitute the values into the point - slope form: $y-(-6)=-4(x - 1)$
Simplify the left - hand side: $y + 6=-4(x - 1)$
Expand the right - hand side: $y+6=-4x + 4$
Subtract 6 from both sides to get the slope - intercept form: $y=-4x+4 - 6$
Simplify the right - hand side: $y=-4x-2$

Step1: Recall slope of perpendicular lines

If two lines are perpendicular, the product of their slopes is $-1$. The given line is $y=-2x + 5$, so its slope $m_1=-2$. Let the slope of the perpendicular line be $m_2$. Then $m_1\times m_2=-1$, so $-2\times m_2=-1$, which gives $m_2=\frac{1}{2}$.

Step2: Use point - slope form to find $b$

The line passes through the point $(-4,10)$ and has a slope $m=\frac{1}{2}$. Using the point - slope form $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(-4,10)$ and $m=\frac{1}{2}$.
Substitute the values: $y - 10=\frac{1}{2}(x+4)$
Expand the right - hand side: $y - 10=\frac{1}{2}x+2$
Add 10 to both sides: $y=\frac{1}{2}x+2 + 10$
Simplify the right - hand side: $y=\frac{1}{2}x + 12$

Answer:

$y = - 4x-2$

Problem 29: