First, establish the slope of a line using the available data before you can derive the equation of a parallel or perpendicular line.

Slope-Intercept Form

y = mx + b

- m is the slope of the line, representing the rate of change between the variables x and y.
- b is the y-intercept, representing the point at which the line crosses the y-axis.
- x and y are the variables that define the points on the line.

## Parallel Lines

What are parallel lines?

Parallel lines are lines in a plane that never cross each other and are always equally spaced apart. The slopes of parallel lines are the same: m_{1} = m_{2}

## Equation of Parallel Lines

**Example 1.** Find an equation of a line parallel to y = 3x+1.

- Identify the slope (m) of the given line.

Given line:

y = 3x + 1

Slope is: m = 3

- Choose a new y-intercept c for the parallel line. c = 2
- Write the equation of the parallel line using the same slope (m) and the new y-intercept (c). y = 3x + 2
- An equation of a line parallel to y = 3x + 1 is y = 3x + 2.

**Example 2. **Find an equation of a line parallel to 3x-4y = 12.

- Rearrange the expression into slope-intercept form.

y = mx + b

-4y = -3x + 12

- Simplify the equation.

- Keep the slope of the equation and choose a new y-intercept. For this example, let us choose (0, 1).

- The equation of a line parallel

**Example 3. **Find the equation of the line parallel to y = 2x+3

and passing through point (2, 3).

- Identify the slope (m) of the given line.

Given line: y = 2x + 3

Slope: m = 2

- Use the point-slope form of a line to write an equation for the parallel line.

y − y_{1} = m(x − x_{1})

Point: (2, 3) | Slope: m = 2

- Plug in the values from the problem into the equation. Solve for y and simplify the equation.

y – 3 = 2(x – 2)

- y – 3 = 2x – 4
- y = 2x – 4 + 3
- y = 2x – 1

4. The equation of the line parallel to y = 2x + 3

passing through point (2, 3) is y = 2x – 1.

## Perpendicular Lines

What are perpendicular lines?

Two lines that form a 90-degree angle at their intersection are said to be perpendicular. In other terms, the lines are orthogonal, which means they make a right angle.

The slope of a perpendicular line in a two-dimensional Cartesian coordinate system is equal to the negative reciprocal of the slope of the primary line:

## Equations Of Perpendicular Lines

**Example 1.** Find the equation of a line perpendicular to y = 2x+1.

- Identify the slope (m) of the line.

Given line: y = 2x + 1

Slope: m = 2

- Find the negative reciprocal of the original line’s slope (m), and use it in the equation of the perpendicular line.

Slope: m = 2

Negative reciprocal: -½

- Write the equation of the perpendicular line using the new slope and any point on the line.

For this example, let’s use the point

(0, 1) as the y-intercept.

The equation of the line perpendicular

**Example 2. **Find the equation of a line perpendicular to 2x-3y = 6.

- Rearrange the equation to slope-intercept form then simplify the equation.

- Find the negative reciprocal of the slope.

- Write the equation of the perpendicular line using the new slope and any point. For this example, let us use (0, 1).

- The line perpendicular

**Example 3.** Find the equation of a line perpendicular to

y = -2x+3 passing through (2, 2).

- Use the slope of the given line, which is -2. Find its negative reciprocal.

Slope: m = -2

Negative reciprocal = ½

- Use the point-slope form of a line using the points given and the negative reciprocal slope. Plug in the values from the problem into the equation and simplify.

- Write the equation of the perpendicular line using the new slope and the point given (2, 2).

- The equation of a line perpendicular

## Using Equations of Parallel & Perpendicular Lines in Real Life

- Architecture & Engineering
- Technology
- Navigation
- Surveying

## Equations of Parallel & Perpendicular Lines & Abstract Thinking

Finding equations of parallel and perpendicular lines is related to abstract thinking in several ways:

- It involves abstract spatial cognition and the ability to visualize lines and their direction in space.
- It includes comprehending the slope, an abstract mathematical notion, and how it relates to the steepness of a line.

- Deriving the equations of the lines involves the ability to work with abstract symbols and algebraic expressions.
- The ability to reason logically and methodically is required to ascertain the abstract connections between the lines.
- It needs the ability to apply abstract mathematical ideas to practical issues, a key component of abstract thinking.

## Quick Review

Utilize the provided data to calculate the slope before attempting to get the line equation. After that, use the slope-intercept form or point-slope form to get the equation by utilizing the slope and a point on the line.

Parallel Lines

- The slope is the same for parallel lines.
- One vertical line runs parallel to another vertical line.

Perpendicular Lines

- The slopes of perpendicular lines are the reciprocals of the opposite:

- A vertical line forms a right angle (90°) with a horizontal line (and vice versa).
- When two lines are perpendicular to each other, they intersect at a 90-degree angle.

## Answer these exercises:

- Find the equation of a line parallel to y = -3x + 4 and passess through (1, -3). Show your solution.
- Find the equation of a line perpendicular to y = -3x + 4 and passes through (5, 7). Show your solution.

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