**Parallel lines in the coordinate plane :**

Parallel lines are two lines that never intersect. In the coordinate plane, that would look like this :

If we take a closer look at these two lines, the slope of both the lines is 1/2.

This can be generalized to any pair of parallel lines. Parallel lines always have the same slope and different *y*−intercepts.

**Postulate (Slopes of parallel lines) : **

In a coordinate plane, two lines are parallel if and only if they have the same slope.

The slope of a non vertical line is the ratio of the vertical change (the rise) to the horizontal change (the run).

If the line passes through the points (x_{1}, y_{1}) and (x_{2}, y_{2}), then the slope is given by

Slope = **rise / run**

**Slope = (y_{2} - y_{1}) / (x_{2} - x_{1})**

Usually, slope is represented by the variable m.

**Example 1 :**

In the diagram given below, find the slope of each line. Determine whether the lines j_{1 }and j_{2 }are parallel.

**Solution : **

Line j_{1} has a slope of

m_{2} = 4/2 = 2

Line j_{2} has a slope of

m_{2} = 2/1 = 2

Since the slope of the lines j_{1 }and j_{2 }are equal, the lines j_{1 }and j_{2 }are parallel.

**Example 2 :**

In the diagram given below, find the slope of each line. Which lines are parallel ?

**Solution : **

**Part 1 : **

Find the slope of the line k_{1}. Line k_{1} is passing through the points (0, 6) and (2, 0).

Let (x_{1}, y_{1}) = (0, 6) and (x_{2}, y_{2}) = (2, 0)

Slope (k_{1}) = (y_{2} - y_{1}) / (x_{2} - x_{1})

Slope (k_{1}) = (0 - 6) / (2 - 0)

Slope (k_{1}) = - 6 / 2

Slope (k_{1}) = - 3

**Part 2 :**

Find the slope of the line k_{2}. Line k_{2} is passing through the points (-2, 6) and (0, 1).

Let (x_{1}, y_{1}) = (-2, 6) and (x_{2}, y_{2}) = (0, 1)

Slope (k_{2}) = (y_{2} - y_{1}) / (x_{2} - x_{1})

Slope (k_{2}) = (1 - 6) / [0 -(-2)]

Slope (k_{2}) = (1 - 6) / [0 + 2]

Slope (k_{2}) = - 5/2

**Part 3 :**

Find the slope of the line k_{3}. Line k_{3} is passing through the points (-6, 5) and (-4, 0).

Let (x_{1}, y_{1}) = (-6, 5) and (x_{2}, y_{2}) = (-4, 0)

Slope (k_{3}) = (y_{2} - y_{1}) / (x_{2} - x_{1})

Slope (k_{3}) = (0 - 5) / [-4 - (-6)]

Slope (k_{3}) = - 5 / (-4 + 6)

Slope (k_{3}) = - 5 / 2

Compare the slopes. Because k_{2} and k_{3} have the same slope, they are parallel. Line k_{1} has a different slope, so it is not parallel to either of the lines.

**Example 3 :**

Write an equation of the line through the point (2, 3) that has a slope 5

**Solution : **

Slope-intercept form equation of a line :

y = mx + b ------(1)

Substitute (x, y) = (2, 3) amd m = 5

3 = 5(2) + b

Simplify

3 = 10 + b

Subtract 3 from both sides

-7 = b

The equation of the required line is

(1) ------> y = 5x - 7

**Example 4 :**

In the diagram given below,

Line n_{1} has the equation y = -x/3 -1.

Line n_{2} is parallel to the line n_{1} and passes through the point (3, 2).

Write the equation of the line n_{2}.

**Solution : **

The slope of the line n_{1} is -1/3. Because the lines n_{1} and n_{2} are parallel, they have the same slope. So, the slope of the line n_{2} is also -1/3.

Slope-intercept form equation of a line :

y = mx + b ------(1)

Because the line n_{2} is passing through (3, 2), substitute aa(x, y) = (3, 2) amd m = -1/3

2 = (-1/3)(3) + b

Simplify

2 = -1 + b

Add 1 to both sides.

3 = b

The equation of the required line is

(1) ------> y = (-1/3)x + 3

y = -x/3 + 3

After having gone through the stuff given above, we hope that the students would have understood "Parallel lines in the coordinate plane".

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