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Finding the equation of a line given the y intercept

The equation of a straight line is usually given in the form $y=mx+c$ where m is the gradient (slope) and c is the y-intercept (where the line cuts the vertical axis).

The gradient = $\frac{rise}{run}$ or $\frac{\delta y}{\delta x}$

Example

The gradient between (2, 5) and (4,10) = $\frac{10-5}{4-2}=\frac{5}{2}$

It is important that if we do 10 – 5 for the rise that we do not do 2 – 4 for the run.  If we are not consistent then we will end up with the wrong sign for the gradient.

Finding the equation given the gradient and the y-intercept

Example

A line has a gradient of 3 and cuts the y axis at (0,4), what is the equation of the line?

From the question we can see that m = 3 and c = 4, so substituting this in to $y=mx+c$ gives $y=3x+4$

Finding the equation given the y-intercept and a parallel line

Example

A line is parallel to the line $y=2x+7$ and intercepts the y axis at (0, -5), what is the equation of the line?

Since the given line has a gradient of 2 (the x coefficient), m = 2.  We can also see that c = -5 from the point.

So substituting this in to $y=mx+c$ gives $y=2x-5$

Finding the equation given the y-intercept and another point

Example

A line passes through the points (0, 4 ) and (3, 11), what is the equation of the line?

We can calculate the gradient using the method shown above so $m=\frac{11-4}{3-0}=\frac{7}{3}$

From the point (0, 4) we can see that c = 4

So substituting this in to $y=mx+c$ gives $y=\frac{7}{3}x+4$