Equation for a straight line

Equation of a straight line

Gradient form

The general equation for a straight line in its gradient form is:

y = mx + c
where = the gradient of the line
c = a constant

If we let x = 0, then it is found that y = c. Therefore is equal to the y-intercept.


Sketch the graph of the equation y = 2x + 3.

Firstly, c = 3 and therefore the y-intercept is at the point (0, 3).
To find the x-intercept, let y = 0. The x-intercept is therefore at (-3/2, 0).
Plot both of these points and join them to create the line.


Intercept form

Another way to express a linear relation is in the form:

ax + by = c
where a, b, and c are all constants

This is referred to as the intercept form as it is convenient in determining the x and y intercepts.


Sketch the graph 8x + 11y = 88.

When x = 0, y = 8. Therefore the y-intercept is at (0, 8).
When y = 0x = 11. Therefore the x-intercept is at (11, 0).


Finding the equation of a straight line

The gradient, m, of the line can be rearranged using a general point (x, y) on the same line

y - y_1 = m(x - x_1)

This form is most convenient when determining the equation of a straight line.

Given two points

When two points of a line are found, they can be substituted into the gradient formula in order to find the slope, m. After that, the re-arranged gradient formula can be used for convenience.

Find the equation of the straight line passing through the points (6, 4) and (2, -4).

The gradient must first be found:
m = \frac{-4 - 4}{2 - 6}
m = 2

This value, as well as any of the given points can be substituted into the formula:
y - 4 = 2(x - (-4))
y = 2x + 12

Given a point, and the gradient

Using the same re-arranged formula, the equation can be quickly deduced.

Find the equation of the line that passes through the point (2,5) and has a gradient of -3. 

Using y - y_1 = m(x - x_1)
y - 5 = -3(x - 2)
y = -3x + 11

Given a graph

In a graph, the value of c reads as the y-intercept. m is the gradient and can be calculated using two points on the line. The equation can therefore be written in the form y = mx + c

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