Parallel Lines in $y = mx + c$

Parallel Lines

• Parallel lines are like railroad tracks: they never meet, no matter how far they extend.
• In the equation $y = mx + c$, parallel lines have identical gradients ($m$) but different y-intercepts ($c$).
• This identical slope means the lines rise over the run at the same rate, keeping them an equal distance apart at all points.

Determining Parallel Lines

Two lines $y = m_1x + c_1$ and $y = m_2x + c_2$ are parallel if $m_1 = m_2$ and $c_1 \neq c_2$. The equality of $m$ ensures the lines have the same steepness, hence never intersecting.

 

Example

If we have the line $y = 2x + 7$, give the equation of the line that would be parallel to this and passes through the point $(3, 8)$.

Solution

Step 1: Determine Gradient 

Gradient of new line is $m_2 = 2$ as the lines are parallel

Step 2: Use the gradient and co-ordinate to solve for c 

$$y = mx + c \to m = 2 \therefore y = 2x + c  \\ \text{Substitute co-ordinate}  8 = 2(3) + c \to 8 = 6 + c \therefore c = 2 \\ y = 2x + 2$$