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Parallel Lines and Slopes

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 ) .