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Gradients of straight-line graphs

The gradient of a line is a measure of how steep the line is.

y

x

a horizontal line

Zero gradient

a downwards slope

Negative gradient

y

x

an upwards slope

Positive gradient

If a line is vertical, its gradient cannot be specified.

y

x

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the gradient =change in ychange in x

Finding the gradient from two given points

If we are given any two points (x1, y1) and (x2, y2) on a line we can calculate the gradient of the line as follows:

Gradient =y2 – y1

x2 – x1

x

y

x2 – x1

(x1, y1)

(x2, y2)

y2 – y1

Draw a right-angled triangle between the two points on the line as follows.

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Calculating gradients

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Investigating linear graphs

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The general equation of a straight line

The general equation of a straight line can be written as:

The value of m tells us the gradient of the line.

The value of c tells us where the line crosses the y-axis.

This is called the y-intercept and it has the coordinate (0, c).

For example, the line y = 3x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4).

y = mx + c

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The gradient and the y-intercept

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Rearranging to y = mx + c

Sometimes the equation of a straight line graph is not given in the form y = mx + c.

Rearrange the equation by performing the same operations on both sides.

2y + x = 4

y = – x + 212

2y = –x + 4subtract x from both sides:

y =–x + 4

2divide both sides by 2:

The equation of a straight line is 2y + x = 4. Find the gradient and the y-intercept of the line.

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Once the equation is in the form y = mx + c we can determine the value of the gradient and the y-intercept.

c = 2 and so the y-intercept is (0, 2).

y = – x + 212

Rearranging to y = mx + c

12

–m = so the gradient of the line is .12

–

x

y

y = – x + 212

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Substituting values into equations

A line with the equation y = mx + 5 passes through the point (3, 11). What is the value of m?

To solve this problem we can substitute x = 3 and y = 11 into the equation y = mx + 5.

This gives us: 11 = 3m + 5

6 = 3msubtract 5 from both sides:

2 = mdivide both sides by 3:

m = 2

The equation of the line is therefore y = 2x + 5.

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What is the equation of the line?

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Match the equations to the graphs