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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 .

Math diagram

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.

$\begin{array}{l} y - 1 + 1 = 2 x - 6 + 1 \\ y = 2 x - 5 \end{array}$

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

Math diagram