What is the slope of the line that is parallel to the given line and passes through the given point?

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How to use Algebra to find parallel and perpendicular lines.

Parallel Lines

How do we know when two lines are parallel?

Their slopes are the same!

The slope is the value m in the equation of a line: y = mx + b

Example:

Find the equation of the line that is:

• parallel to y = 2x + 1
• and passes though the point (5,4)

The slope of y=2x+1 is: 2

The parallel line needs to have the same slope of 2.

We can solve it using the "point-slope" equation of a line:

y − y1 = 2(x − x1)

And then put in the point (5,4):

y − 4 = 2(x − 5)

And that answer is OK, but let's also put it in y = mx + b form:

y − 4 = 2x − 10

y = 2x − 6

Vertical Lines

But this does not work for vertical lines ... I explain why at the end.

Not The Same Line

Be careful! They may be the same line (but with a different equation), and so are not parallel.

How do we know if they are really the same line? Check their y-intercepts (where they cross the y-axis) as well as their slope:

For y = 3x + 2: the slope is 3, and y-intercept is 2

For y − 2 = 3x: the slope is 3, and y-intercept is 2

In fact they are the same line and so are not parallel

Perpendicular Lines

Two lines are Perpendicular when they meet at a right angle (90°).

To find a perpendicular slope:

When one line has a slope of m, a perpendicular line has a slope of −1m

In other words the negative reciprocal

Example:

Find the equation of the line that is

• perpendicular to y = −4x + 10
• and passes though the point (7,2)

The slope of y=−4x+10 is: −4

The negative reciprocal of that slope is:

m = −1−4 = 14

So the perpendicular line will have a slope of 1/4:

y − y1 = (1/4)(x − x1)

And now put in the point (7,2):

y − 2 = (1/4)(x − 7)

And that answer is OK, but let's also put it in "y=mx+b" form:

y − 2 = x/4 − 7/4

y = x/4 + 1/4

Quick Check of Perpendicular

When we multiply a slope m by its perpendicular slope −1m we get simply −1.

So to quickly check if two lines are perpendicular:

When we multiply their slopes, we get −1

Like this:

Are these two lines perpendicular?

When we multiply the two slopes we get:

2 × (−0.5) = −1

Yes, we got −1, so they are perpendicular.

Vertical Lines

The previous methods work nicely except for a vertical line:

In this case the gradient is undefined (as we cannot divide by 0):

m = yA − yBxA − xB = 4 − 12 − 2 = 30 = undefined

So just rely on the fact that:

• a vertical line is parallel to another vertical line.
• a vertical line is perpendicular to a horizontal line (and vice versa).

Summary

• parallel lines: same slope
• perpendicular lines: negative reciprocal slope (−1/m)

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The calculator will find the equation of the parallel/perpendicular line to the given line passing through the given point, with steps shown.

For drawing lines, use the graphing calculator.

Solution

Your input: find the equation of the line parallel to the line $$$y=2 x + 5$$$ passing through the point $$$\left(-3,5\right)$$$.

The equation of the line in the slope-intercept form is $$$y=2 x + 5$$$.

The slope of the parallel line is the same: $$$m=2$$$.

So, the equation of the parallel line is $$$y=2 x+a$$$.

To find $$$a$$$, we use the fact that the line should pass through the given point: $$$5=\left(2\right) \cdot \left(-3\right)+a$$$.

Thus, $$$a=11$$$.

Therefore, the equation of the line is $$$y=2 x + 11$$$.

Answer: $$$y=2 x + 11$$$.

Parallel lines are coplanar lines that do not intersect. In two dimensions, parallel lines have the same slope .

We can write the equation of a line parallel to a given line if we know a point on the line and an equation of the given line.

Example:

Write the equation of a line that passes through the point ( 3 , 1 ) and is parallel to the line

y = 2 x + 3 .

Parallel lines have the same slope.

The slope of the line with equation y = 2 x + 3 is 2 . So, any line parallel to y = 2 x + 3 has the same slope 2 .

Now use the point-slope form to find the equation.

y − y 1 = m ( x − x 1 )

We have to find the equation of the line which has slope 2 and passes through the point ( 3 , 1 ) . So, replace m with 2 , x 1 with 3 , and y 1 with 1 .

y − 1 = 2 ( x − 3 )

Use the distributive property .

y − 1 = 2 x − 6

Add 1 to each side.

y − 1 + 1 = 2 x − 6 + 1                           y = 2 x − 5

Therefore, the line y = 2 x − 5 is parallel to the line y = 2 x + 3 and passes through the point ( 3 , 1 ) .