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Slope of Parallel Lines

Parallel lines are the lines which does not intersect in any direction. Slope of a line is the value of the angle that a straight line makes with the positive direction of x-axis in the anticlockwise sense. Basically for any straight line y = mx + b, m is referred to as slope. 

Slopes of two parallel lines are equal. If the slope of two parallel lines are $m_1$ and  $m_2$ then $m_1$ = $m_2$.

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Slope of Parallel Lines Definition

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Slope of a line in ratio of change in y axis to x axis that is the small change in entity in the y axis to the small change in entity in the x axis is called as slope. Parallel lines are the lines that lie in the same plane have the property that they never intersect each other. They never meet at a point. They are always the same distance apart.

Let y = m1x + c1 and y = m2x + c2 be two parallel lines then there slopes will be equal.

=> m1 = m2

Slope of Two Parallel Lines

In the above diagram, we can see the slope of two parallel lines. Both the slopes will have same value. Slope is the ratio of change in y to change in x. When we consider two parallel lines there ratios will be equivalent.

Slope of Parallel Lines Property

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Parallel lines have the value of the angle made with the x axis always same or always have the same slope.

Let the equation of two lines be y = (m1)x + c1 and y = (m2)x + c2 where, m1 and m2 are slopes of the lines. The two lines are parallel if and only if m1 = m2.

Example on Slope of Two Parallel Lines


Parallel Lines Slope Intercept Form

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Two lines are parallel if they have same slope. The equation of the parallel line in slope-intercept form is y = mx + c, where m is the slope of the line.

Intercept Theorem of Parallel Lines

If a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.

Slope Intercept Theorem of Parallel Lines

Data:

AP || BQ || CR. The intercepts AB and BC are equal. PQ, QR are the intercepts on any other line.

To Prove Intercept Theorem:

PQ = QR

Construction:

Through A and B draw AE and BF parallel to PQR to cut BQ and CR at E and F respectively.

To prove that PQ = QR

Proof:

Slope Intercept Theorem of Parallel Lines

Slope of Parallel Lines Examples

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Given below are some of the examples on slope of parallel lines.

Solved Examples

Question 1: Find the equation of a straight line parallel to y-axis and passing through the point (-3, 2).
Solution:

We know that the equation of a straight line parallel to y-axis is

x = a

Since it passes through the point (-3, 2), we get -3 = a i.e.

a = -3.

Substituting this values of a, we get

x = - 3 i.e. x + 3 = 0, which is the required equation.



Question 2: Given the equations of two lines 2y - 4x = 12 and y = 2x + 8. Check whether the given lines are parallel or not?
Solution:

In order to check that the lines are parallel or not, we have to calculate slope. First, convert the equation of lines into standard form y = mx + b to identify the slope.

Given the equation of the first line is

2y - 4x = 12

2y = 4x + 12

Dividing both sides by 2, we have

y = 2x + 6

Hence, slope of the first line is

m1 = 2.

Given the equation of the second line is

y = 2x + 8. It is already in the standard form.

Hence, slope of the second line is

m2 = 2.

Since the value of the slopes m1 = m2 = 2 are equal.

Hence, the given lines are parallel.(Answer)



Question 3: Given the equation of a line y = 5x + 8. What is the slope of the line parallel to given line?
Solution:

The equation of the line given is y = 5x + 8

It is in the standard form y = mx + b.

Hence, slope of the line is 5.

As we have to find the slope of the line parallel to the line y = 5x + 8.

We know that, the slopes of the two parallel lines must be the same.

Hence, the slope of required line is = 5 (Answer)



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