Question

determine whether the following are parallel, perpendicular, or neither.

1. $y = 5x - 10$ $m=\underline{\quad}$

$2y - 10x = -6$ $m=\underline{\quad}$

1. $-15x + 5y = 15$ $m=\underline{\quad}$

$y = 1 - 3x$ $m=\underline{\quad}$
write an equation of a line in slope - intercept form that passes through the given point and is parallel to the given line.

1. $x + 2y = -10$ $(2, 1)$

write an equation of a line in slope - intercept form that passes through the given point and is perpendicular to the given line.

1. $8x + 5y = -25$ $(-5, 3)$

Explanation:

Problem 1: Determine Slopes and Relationship

Step 1: Find slope of \( y = 5x - 10 \)

The equation is in slope - intercept form \( y=mx + b \), where \( m \) is the slope. So for \( y = 5x-10 \), \( m = 5 \).

Step 2: Find slope of \( 2y-10x=-6 \)

First, we rewrite the equation in slope - intercept form.
Add \( 10x \) to both sides: \( 2y=10x - 6 \)
Divide both sides by 2: \( y = 5x-3 \)
So the slope \( m = 5 \) (not - 5 as in the original wrong calculation).
Since the slopes of the two lines \( y = 5x - 10 \) and \( y=5x - 3 \) are equal (\( m_1=m_2 = 5 \)), the lines are parallel.

Problem 2: Determine Slopes and Relationship

Step 1: Find slope of \( - 15x + 5y=15 \)

Rewrite the equation in slope - intercept form.
Add \( 15x \) to both sides: \( 5y=15x + 15 \)
Divide both sides by 5: \( y = 3x+3 \)
So the slope \( m = 3 \).

Step 2: Find slope of \( y = 1-3x \)

The equation is in slope - intercept form \( y=mx + b \), with \( m=-3 \).
The product of the slopes \( 3\times(-3)=-9
eq - 1 \) and \( 3
eq - 3 \), so the lines are neither parallel nor perpendicular.

Problem 3: Equation of Parallel Line

Step 1: Find slope of \( x + 2y=-10 \)

Rewrite in slope - intercept form.
Subtract \( x \) from both sides: \( 2y=-x - 10 \)
Divide by 2: \( y=-\frac{1}{2}x-5 \)
The slope of the given line is \( m =-\frac{1}{2} \). Since parallel lines have the same slope, the slope of the line we want to find is also \( m =-\frac{1}{2} \).

Step 2: Use point - slope form \( y - y_1=m(x - x_1) \)

We have the point \( (x_1,y_1)=(2,1) \) and \( m =-\frac{1}{2} \).
\( y - 1=-\frac{1}{2}(x - 2) \)
Expand: \( y-1=-\frac{1}{2}x + 1 \)
Add 1 to both sides: \( y=-\frac{1}{2}x+2 \)

Problem 4: Equation of Perpendicular Line

Step 1: Find slope of \( 8x + 5y=-25 \)

Rewrite in slope - intercept form.
Subtract \( 8x \) from both sides: \( 5y=-8x - 25 \)
Divide by 5: \( y=-\frac{8}{5}x-5 \)
The slope of the given line is \( m_1=-\frac{8}{5} \). The slope of a line perpendicular to a line with slope \( m_1 \) is \( m_2=\frac{5}{8} \) (since \( m_1\times m_2=-1\), so \( m_2=\frac{-1}{m_1}=\frac{5}{8} \)).

Step 2: Use point - slope form \( y - y_1=m(x - x_1) \)

We have the point \( (x_1,y_1)=(-5,3) \) and \( m=\frac{5}{8} \).
\( y - 3=\frac{5}{8}(x + 5) \)
Expand: \( y-3=\frac{5}{8}x+\frac{25}{8} \)
Add 3 (which is \( \frac{24}{8} \)) to both sides: \( y=\frac{5}{8}x+\frac{25 + 24}{8}=\frac{5}{8}x+\frac{49}{8} \)

Final Answers

1. Slope of \( y = 5x-10 \) is \( \boldsymbol{5} \), slope of \( 2y - 10x=-6 \) is \( \boldsymbol{5} \), lines are \(\boldsymbol{\text{parallel}}\).
2. Slope of \( - 15x + 5y = 15 \) is \( \boldsymbol{3} \), slope of \( y=1 - 3x \) is \( \boldsymbol{-3} \), lines are \(\boldsymbol{\text{neither}}\).
3. Equation of the parallel line: \( \boldsymbol{y=-\frac{1}{2}x + 2} \)
4. Equation of the perpendicular line: \( \boldsymbol{y=\frac{5}{8}x+\frac{49}{8}} \)

Answer:

Problem 1: Determine Slopes and Relationship

Step 1: Find slope of \( y = 5x - 10 \)

The equation is in slope - intercept form \( y=mx + b \), where \( m \) is the slope. So for \( y = 5x-10 \), \( m = 5 \).

Step 2: Find slope of \( 2y-10x=-6 \)

First, we rewrite the equation in slope - intercept form.
Add \( 10x \) to both sides: \( 2y=10x - 6 \)
Divide both sides by 2: \( y = 5x-3 \)
So the slope \( m = 5 \) (not - 5 as in the original wrong calculation).
Since the slopes of the two lines \( y = 5x - 10 \) and \( y=5x - 3 \) are equal (\( m_1=m_2 = 5 \)), the lines are parallel.

Problem 2: Determine Slopes and Relationship

Step 1: Find slope of \( - 15x + 5y=15 \)

Rewrite the equation in slope - intercept form.
Add \( 15x \) to both sides: \( 5y=15x + 15 \)
Divide both sides by 5: \( y = 3x+3 \)
So the slope \( m = 3 \).

Step 2: Find slope of \( y = 1-3x \)

The equation is in slope - intercept form \( y=mx + b \), with \( m=-3 \).
The product of the slopes \( 3\times(-3)=-9
eq - 1 \) and \( 3
eq - 3 \), so the lines are neither parallel nor perpendicular.

Problem 3: Equation of Parallel Line

Step 1: Find slope of \( x + 2y=-10 \)

Rewrite in slope - intercept form.
Subtract \( x \) from both sides: \( 2y=-x - 10 \)
Divide by 2: \( y=-\frac{1}{2}x-5 \)
The slope of the given line is \( m =-\frac{1}{2} \). Since parallel lines have the same slope, the slope of the line we want to find is also \( m =-\frac{1}{2} \).

Step 2: Use point - slope form \( y - y_1=m(x - x_1) \)

We have the point \( (x_1,y_1)=(2,1) \) and \( m =-\frac{1}{2} \).
\( y - 1=-\frac{1}{2}(x - 2) \)
Expand: \( y-1=-\frac{1}{2}x + 1 \)
Add 1 to both sides: \( y=-\frac{1}{2}x+2 \)

Problem 4: Equation of Perpendicular Line

Step 1: Find slope of \( 8x + 5y=-25 \)

Rewrite in slope - intercept form.
Subtract \( 8x \) from both sides: \( 5y=-8x - 25 \)
Divide by 5: \( y=-\frac{8}{5}x-5 \)
The slope of the given line is \( m_1=-\frac{8}{5} \). The slope of a line perpendicular to a line with slope \( m_1 \) is \( m_2=\frac{5}{8} \) (since \( m_1\times m_2=-1\), so \( m_2=\frac{-1}{m_1}=\frac{5}{8} \)).

Step 2: Use point - slope form \( y - y_1=m(x - x_1) \)

We have the point \( (x_1,y_1)=(-5,3) \) and \( m=\frac{5}{8} \).
\( y - 3=\frac{5}{8}(x + 5) \)
Expand: \( y-3=\frac{5}{8}x+\frac{25}{8} \)
Add 3 (which is \( \frac{24}{8} \)) to both sides: \( y=\frac{5}{8}x+\frac{25 + 24}{8}=\frac{5}{8}x+\frac{49}{8} \)

Final Answers

1. Slope of \( y = 5x-10 \) is \( \boldsymbol{5} \), slope of \( 2y - 10x=-6 \) is \( \boldsymbol{5} \), lines are \(\boldsymbol{\text{parallel}}\).
2. Slope of \( - 15x + 5y = 15 \) is \( \boldsymbol{3} \), slope of \( y=1 - 3x \) is \( \boldsymbol{-3} \), lines are \(\boldsymbol{\text{neither}}\).
3. Equation of the parallel line: \( \boldsymbol{y=-\frac{1}{2}x + 2} \)
4. Equation of the perpendicular line: \( \boldsymbol{y=\frac{5}{8}x+\frac{49}{8}} \)