Linear Equations -Writing Equations of Perpendicular Lines ... - Free Printable

The task involves finding the equation of a line that is perpendicular to a given line and passes through a specific point. To solve these problems, we need to follow these steps:

Steps to Solve:

1. Find the slope of the given line: The slope of a line in the form \( y = mx + b \) is \( m \).
2. Determine the slope of the perpendicular line: The slope of a line perpendicular to another is the negative reciprocal of the original slope. If the slope of the given line is \( m \), then the slope of the perpendicular line is \( -\frac{1}{m} \).
3. Use the point-slope form of the line equation: The point-slope form is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the given point and \( m \) is the slope of the perpendicular line.
4. Simplify the equation: Convert the equation into the slope-intercept form \( y = mx + b \).

Let's solve each problem step by step.

---

Problem 1: \( y = 2x - 3 \); Point: (2, 5)

1. Slope of the given line: The slope \( m \) of \( y = 2x - 3 \) is \( 2 \).
2. Slope of the perpendicular line: The negative reciprocal of \( 2 \) is \( -\frac{1}{2} \).
3. Point-slope form: Using the point (2, 5) and slope \( -\frac{1}{2} \):
\[
y - 5 = -\frac{1}{2}(x - 2)
\]
4. Simplify:
\[
y - 5 = -\frac{1}{2}x + 1
\]
\[
y = -\frac{1}{2}x + 6
\]

Answer: \( y = -\frac{1}{2}x + 6 \)

---

Problem 2: \( y = -x + 8 \); Point: (9, 2)

1. Slope of the given line: The slope \( m \) of \( y = -x + 8 \) is \( -1 \).
2. Slope of the perpendicular line: The negative reciprocal of \( -1 \) is \( 1 \).
3. Point-slope form: Using the point (9, 2) and slope \( 1 \):
\[
y - 2 = 1(x - 9)
\]
4. Simplify:
\[
y - 2 = x - 9
\]
\[
y = x - 7
\]

Answer: \( y = x - 7 \)

---

Problem 3: \( y = \frac{1}{3}x + 6 \); Point: (0, -3)

1. Slope of the given line: The slope \( m \) of \( y = \frac{1}{3}x + 6 \) is \( \frac{1}{3} \).
2. Slope of the perpendicular line: The negative reciprocal of \( \frac{1}{3} \) is \( -3 \).
3. Point-slope form: Using the point (0, -3) and slope \( -3 \):
\[
y - (-3) = -3(x - 0)
\]
4. Simplify:
\[
y + 3 = -3x
\]
\[
y = -3x - 3
\]

Answer: \( y = -3x - 3 \)

---

Problem 4: \( y = -\frac{1}{2}x + 3 \); Point: (4, -2)

1. Slope of the given line: The slope \( m \) of \( y = -\frac{1}{2}x + 3 \) is \( -\frac{1}{2} \).
2. Slope of the perpendicular line: The negative reciprocal of \( -\frac{1}{2} \) is \( 2 \).
3. Point-slope form: Using the point (4, -2) and slope \( 2 \):
\[
y - (-2) = 2(x - 4)
\]
4. Simplify:
\[
y + 2 = 2x - 8
\]
\[
y = 2x - 10
\]

Answer: \( y = 2x - 10 \)

---

Problem 5: \( y = 3x - 4 \); Point: (1, -1)

1. Slope of the given line: The slope \( m \) of \( y = 3x - 4 \) is \( 3 \).
2. Slope of the perpendicular line: The negative reciprocal of \( 3 \) is \( -\frac{1}{3} \).
3. Point-slope form: Using the point (1, -1) and slope \( -\frac{1}{3} \):
\[
y - (-1) = -\frac{1}{3}(x - 1)
\]
4. Simplify:
\[
y + 1 = -\frac{1}{3}x + \frac{1}{3}
\]
\[
y = -\frac{1}{3}x + \frac{1}{3} - 1
\]
\[
y = -\frac{1}{3}x - \frac{2}{3}
\]

Answer: \( y = -\frac{1}{3}x - \frac{2}{3} \)

---

Problem 6: \( y = \frac{2}{3}x - 2 \); Point: (2, 1)

1. Slope of the given line: The slope \( m \) of \( y = \frac{2}{3}x - 2 \) is \( \frac{2}{3} \).
2. Slope of the perpendicular line: The negative reciprocal of \( \frac{2}{3} \) is \( -\frac{3}{2} \).
3. Point-slope form: Using the point (2, 1) and slope \( -\frac{3}{2} \):
\[
y - 1 = -\frac{3}{2}(x - 2)
\]
4. Simplify:
\[
y - 1 = -\frac{3}{2}x + 3
\]
\[
y = -\frac{3}{2}x + 4
\]

Answer: \( y = -\frac{3}{2}x + 4 \)

---

Problem 7: \( y = -\frac{1}{4}x + 5 \); Point: (4, 3)

1. Slope of the given line: The slope \( m \) of \( y = -\frac{1}{4}x + 5 \) is \( -\frac{1}{4} \).
2. Slope of the perpendicular line: The negative reciprocal of \( -\frac{1}{4} \) is \( 4 \).
3. Point-slope form: Using the point (4, 3) and slope \( 4 \):
\[
y - 3 = 4(x - 4)
\]
4. Simplify:
\[
y - 3 = 4x - 16
\]
\[
y = 4x - 13
\]

Answer: \( y = 4x - 13 \)

---

Problem 8: \( y = \frac{3}{4}x + 2 \); Point: (0, -2)

1. Slope of the given line: The slope \( m \) of \( y = \frac{3}{4}x + 2 \) is \( \frac{3}{4} \).
2. Slope of the perpendicular line: The negative reciprocal of \( \frac{3}{4} \) is \( -\frac{4}{3} \).
3. Point-slope form: Using the point (0, -2) and slope \( -\frac{4}{3} \):
\[
y - (-2) = -\frac{4}{3}(x - 0)
\]
4. Simplify:
\[
y + 2 = -\frac{4}{3}x
\]
\[
y = -\frac{4}{3}x - 2
\]

Answer: \( y = -\frac{4}{3}x - 2 \)

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ y = -\frac{1}{2}x + 6 \\
2. & \ y = x - 7 \\
3. & \ y = -3x - 3 \\
4. & \ y = 2x - 10 \\
5. & \ y = -\frac{1}{3}x - \frac{2}{3} \\
6. & \ y = -\frac{3}{2}x + 4 \\
7. & \ y = 4x - 13 \\
8. & \ y = -\frac{4}{3}x - 2
\end{aligned}
}
\]