Parallel lines are two lines on a given plane that never cross (meaning they will go on forever without touching). An important feature of parallel lines is that they both have the same slope. Slope can be defined as the elevation (change in X coordinates) of a line or, in other words, its angulation. Parallel lines are most commonly represented by two vertical lines (ll). For example, ABllCD indicates that AB are parallel to CD.

## Steps

### Method 1 of 3: Comparing the Slopes of Each Row

#### Step 1. Define the slope formula

The slope of a line is defined as (Y_{2} - Y_{1})/(X_{2} - X_{1}), where X and Y represent the horizontal and vertical coordinates of points on it. To calculate this formula, you must define two points. The one closest to the base of the line will be (X_{1}, X_{1}) and the highest will be (X_{2}, X_{2}).

- This formula can also be called the slope of the line. It represents the vertical difference over the horizontal, or its slope.
- If a line is facing up and to the right, it has a positive slope.
- If the line is facing down and to the right, it has a negative slope.

#### Step 2. Identify the X and Y coordinates of two points present on each line

A point on a line is given by coordinates (X, Y), where X represents the location on the horizontal axis and Y the location on the vertical axis. To calculate the slope, you must identify two points on each of the lines under study.

- These points can be easily determined if the line is drawn on graph paper.
- To determine a point, draw a dotted line from the horizontal axis until it crosses the original line. The starting position on the horizontal axis represents the X coordinate while the Y will be the point where the dotted line intersects the vertical axis.
- For example, line l has points (1, 5) and (-2, 4), while line r has points (3, 3) and (1, -4).

#### Step 3. Enter the points for each line in the slope formula

To calculate the slope, just enter the numbers and perform the respective subtraction and division. Put the determined coordinates into the X and Y values of the formula.

- To calculate the slope of line l: slope = (5 - (-4))/(1 - (-2))
- Subtraction: slope = 9/3
- Division: slope = 3
- The slope of the r line is: slope = (3 - (-4))/(3 - 1) = 7/2

#### Step 4. Compare the slopes of each row

Remember that two lines are only parallel if they have identical slopes. They can look parallel on paper and even be quite close together - however, if they don't have exactly the same slopes, they aren't parallel.

### In this example, 3 is not equal to 7/2, so these lines are not parallel

### Method 2 of 3: Using the Line Equation

#### Step 1. Determine the equation of the straight line

The straight line equation has the basic formula y = mx + b, where m represents the slope, b represents the y axis, and x and y are variables representing coordinates on the line - generally, they remain as x and y in the equation. In this format, you can easily determine the slope of the line as the variable "m".

### For example, rewrite 4y - 12x = 20 and y = 3x - 1. The equation 4y - 12x = 20 must be rewritten algebraically, while y = 3x - 1 is already in the basic formula of the line equation and does not need to be reordered

#### Step 2. Rewrite the formula as an equation of the line

Sometimes the formula for the line is not yet ordered as an equation for the line. All it takes is a little math and effort to rearrange the variables and get the desired format.

- For example: rewrite the line 4y - 12x = 20 as the equation of the line.
- Add 12x to both sides of the equation: 4y - 12x + 12x = 20 + 12x.
- Divide each side by 4 to get the result of y: 4y/4 = 12x/4 + 20/4.
- Line equation: y = 3x + 5.

#### Step 3. Compare the slopes of each line

Remember that when two lines are parallel to each other, they will both have the same slope. With the equation y = mx + b, where m represents the slope of the line, you can identify and compare the slope of each of them.

- In our example, the first line has the formula y = 3x + 5, so its slope is equal to 3. The other line has the formula y = 3x - 1, also with a slope equal to 3. Like both slopes are identical, it means the two lines are parallel.
- Note that if these equations had the same Y value, they would both be a single line rather than just parallel.

### Method 3 of 3: Using a Point and Slope

#### Step 1. Use the point and slope method

This form allows you to write the equation of the line if you know its slope and have an (x, y) coordinate. It can be used if you want to determine a second line parallel to an existing one with a defined slope. The formula is y - y_{1} = m(x - x_{1}), where m represents the slope of the line, x_{1} represents the x coordinate of a point on the line and y_{1} represents the y coordinate of the same point. As with the previous method, x and y are variables representing coordinates present in the line - they will generally remain as x and y in the equation.

### The following steps work in this example: Write the equation of a line parallel to the line y = -4x + 3 that passes through the point (1, -2)

#### Step 2. Determine the slope of the first row

When writing the formula for a new line, you must first identify the slope of the existing one. It is important that, for the original line, you use the equation of the straight line and know its respective slope (m).

### The original line can be represented by y = -4x + 3. In this equation, -4 represents the variable m and thus the slope of the line

#### Step 3. Identify a point on the new line

This equation works only if you have a coordinate that goes through the new line. Remember to choose one that is not already present in the original line. If the final formulas have the same equation as the line, they are not parallel, but the same line.

### In our example, we will use the coordinate (1, -2)

#### Step 4. Write the formula for the new line with the equation for the line

Remember the formula is y - y_{1} = m(x - x_{1}). Enter the slope and coordinates of the point to write the formula for the new line that will be parallel to the first.

### In our example, with slope (m) equal to -4 and coordinates (x, y) equal to (1, -2): y - (-2) = -4(x - 1)

#### Step 5. Simplify the equation

After entering the numbers, the equation should be simplified to its most common form. This line of the equation, if projected onto a Cartesian plane, will be parallel to the original equation.

- For example: y - (-2) = -4(x - 1)
- Two negatives form a positive: y + 2 = -4(x - 1)
- Distribute the -4 to x and -1: y + 2 = -4x + 4.
- Subtract -2 from both sides: y + 2 - 2 = -4x + 4 - 2.
- Simplified equation: y = -4x + 2.