Gradient (also called *slope*) is the steepness of a line. To determine the gradient, observe the line from left to right: if the line moves up then the gradient is positive, but if the direction is down then the gradient is negative.

The most basic formula for gradient is:

Try the following Geogebra to get a grasp of gradient movements.

### Gradient from Linear Equation

The gradient-intercept formula is in the form of:

where m is the gradient and c is the y-intercept.

### General Form

Sometimes, the general formula is written instead of the gradient-intercept form. Remember that:

Then, the gradient is given by:

Proof:

#### Example:

Find the gradient of .

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### Equation of a Straight Line

**Point-Intercept Formula**

The equation of a straight line is given by:

To find out the equation of a straight line given the gradient and a point, then use the point-intercept formula.

#### Example:

Find the equation of a line that has a gradient of 3 and passes through the point (1, 2).

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To double check that the line y=3x-1 correctly passes the point (1, 2), substitute x=1 into the equation and check if y=2.

**Two-Point Formula**

In the event that two points are given to find the equation of a line, the two-point formula is needed:

#### Example:

Find the equation of a line that passes through the points (2,3) and (4,7).

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### Parallel and Perpendicular Lines

If two lines are parallel, then they have the same gradient.

#### Example:

Find the equation of a straight line parallel to the line 2x-y-3=0 and passes through (1, -5).

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This means that equal gradient lines would not meet because they are on the same direction.

If two lines are perpendicular, then there is a characteristic determined by the gradients of the two lines.

#### Example:

Show that the lines and are perpendicular.

### Show Answer

Therefore, the lines are perpendicular.