How to find the equation of a linear graph (straight line graph) if its drawn on a coordinate grid.

The equation of a linear graph (a straight line) takes the form of y = mx + c, where m is the gradient of the line and c is the intercept.

To work out m all you need to do is pick two coordinate points the line is passing through, and turn these two points into a right angled triangle. Once this is done, divide the vertical height of the triangle by the horizontal distance (up ÷ across). It doesn’t matter which two points you choose on your line as the gradient will always come out as the same answer.  Also, you need to decide if the line is a positive or negative gradient. If the line is going up from left to right then the gradient is positive and if it’s going downwards from left to right then the gradient is negative.

The next thing to do is work out c. This is really easy, as this is the number that the line passes through on the y axis.

Once this is done the equation of the line can be written down by subbing in the values you have just found into y = mx + c.

Example 1

Find the equation of this linear graph.

First find m.

Let’s pick (2,1) and (4,7) and make a right angled triangle.

The up distance of the triangle is 6 and the horizontal distance of the triangle is 2.

So 6 ÷ 2 = 3

Also m is positive as the graph is going up from left to right.

So m = 3

Next, work out the value of c.

c = -5, as the graph is crossing the y axis at -5 on the y axis.

Finally, sub m = 3 and c = -5 into y = mx + c.

So the linear graph has equation y = 3x – 5

Example 2

Find the equation of this linear graph.

First find m.

Let’s pick (-6,5) and (3,2) and make a right angled triangle.

The up distance of the triangle is 6 and the horizontal distance of the triangle is 2.

So 3 ÷ 9 =1/3 (or 0.3)

Also m is negative as the graph is going down from left to right.

So m = -⅓

Next, work out the value of c.

c = 3, as the graph is crossing the y axis at 3 on the y axis.

Finally, sub m = -⅓ and c = 3 into y = mx + c.

So the linear graph has equation y = -⅓x + 3

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