Worksheet for practicing writing equations of parallel and perpendicular lines with given points and slopes.

To solve the problems, we need to use the concepts of parallel and perpendicular lines. Here's a step-by-step explanation for each part:

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1. Parallel to \( y = -5x + 2 \) and passes through point \( (1, -4) \)

#### Step 1: Identify the slope of the given line
The given line is \( y = -5x + 2 \). The slope \( m \) of this line is \( -5 \).

#### Step 2: Use the slope for the parallel line
Parallel lines have the same slope. Therefore, the slope \( m \) of the new line is also \( -5 \).

#### Step 3: Use the point-slope form to find the equation
The point-slope form of a line is:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is the given point and \( m \) is the slope.

Substitute \( m = -5 \) and the point \( (1, -4) \):
\[
y - (-4) = -5(x - 1)
\]
Simplify:
\[
y + 4 = -5(x - 1)
\]
\[
y + 4 = -5x + 5
\]
\[
y = -5x + 5 - 4
\]
\[
y = -5x + 1
\]

#### Final Answer:
\[
\boxed{y = -5x + 1}
\]

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2. Parallel to \( y = -x - 5 \) and passes through point \( (1, -3) \)

#### Step 1: Identify the slope of the given line
The given line is \( y = -x - 5 \). The slope \( m \) of this line is \( -1 \).

#### Step 2: Use the slope for the parallel line
Parallel lines have the same slope. Therefore, the slope \( m \) of the new line is also \( -1 \).

#### Step 3: Use the point-slope form to find the equation
Substitute \( m = -1 \) and the point \( (1, -3) \):
\[
y - (-3) = -1(x - 1)
\]
Simplify:
\[
y + 3 = -1(x - 1)
\]
\[
y + 3 = -x + 1
\]
\[
y = -x + 1 - 3
\]
\[
y = -x - 2
\]

#### Final Answer:
\[
\boxed{y = -x - 2}
\]

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3. Perpendicular to \( y = \frac{1}{3}x + 3 \) and passes through point \( (2, -4) \)

#### Step 1: Identify the slope of the given line
The given line is \( y = \frac{1}{3}x + 3 \). The slope \( m \) of this line is \( \frac{1}{3} \).

#### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of \( \frac{1}{3} \) is \( -3 \). Therefore, the slope \( m \) of the new line is \( -3 \).

#### Step 3: Use the point-slope form to find the equation
Substitute \( m = -3 \) and the point \( (2, -4) \):
\[
y - (-4) = -3(x - 2)
\]
Simplify:
\[
y + 4 = -3(x - 2)
\]
\[
y + 4 = -3x + 6
\]
\[
y = -3x + 6 - 4
\]
\[
y = -3x + 2
\]

#### Final Answer:
\[
\boxed{y = -3x + 2}
\]

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4. Perpendicular to \( y = -\frac{1}{2}x \) and passes through point \( (4, 3) \)

#### Step 1: Identify the slope of the given line
The given line is \( y = -\frac{1}{2}x \). The slope \( m \) of this line is \( -\frac{1}{2} \).

#### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of \( -\frac{1}{2} \) is \( 2 \). Therefore, the slope \( m \) of the new line is \( 2 \).

#### Step 3: Use the point-slope form to find the equation
Substitute \( m = 2 \) and the point \( (4, 3) \):
\[
y - 3 = 2(x - 4)
\]
Simplify:
\[
y - 3 = 2(x - 4)
\]
\[
y - 3 = 2x - 8
\]
\[
y = 2x - 8 + 3
\]
\[
y = 2x - 5
\]

#### Final Answer:
\[
\boxed{y = 2x - 5}
\]

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Summary of Answers:

1. Parallel to \( y = -5x + 2 \) and passes through \( (1, -4) \): \(\boxed{y = -5x + 1}\)
2. Parallel to \( y = -x - 5 \) and passes through \( (1, -3) \): \(\boxed{y = -x - 2}\)
3. Perpendicular to \( y = \frac{1}{3}x + 3 \) and passes through \( (2, -4) \): \(\boxed{y = -3x + 2}\)
4. Perpendicular to \( y = -\frac{1}{2}x \) and passes through \( (4, 3) \): \(\boxed{y = 2x - 5}\)