Understanding Gradients

The gradient of a straight line tells you how steep the line is.

In the equation \( y = mx + c \), the gradient is \( m \).

 

The gradient is the key feature used to identify whether lines are parallel or perpendicular.

 

 

Parallel Lines

Parallel lines have the same gradient.

 

If a given line has the equation

$$
y = mx + c
$$

then any line parallel to it will also have gradient \( m \) but a different value of \( c \).

 

Example

The line

$$
y = 3x + 1
$$

has gradient \( 3 \).

 

A line parallel to it could be

$$
y = 3x - 4
$$

 

Because the gradients are the same, the lines are parallel and will never meet.

 

 

Perpendicular Lines

Perpendicular lines meet at a right angle.

 

Their gradients multiply to give \( -1 \).

 

If a line has gradient \( m \), the gradient of a line perpendicular to it is

$$
-\frac{1}{m}
$$

 

This is called the negative reciprocal.

 

Example

The line

$$
y = 2x + 5
$$

has gradient \( 2 \).

 

A perpendicular line has gradient

$$
-\frac{1}{2}
$$

 

So a perpendicular line could be

$$
y = -\frac{1}{2}x + 3
$$

 

 

Finding the Equation of a Required Line

To find the equation of a line that is parallel or perpendicular to another line, first identify the correct gradient.

Then substitute the gradient and a known point into \( y = mx + c \) to find \( c \).

 

Example

Find the equation of the line perpendicular to

$$
y = -4x + 7
$$

that passes through the point \( (2, 1) \).

 

The original gradient is \( -4 \).

 

The perpendicular gradient is

$$
\frac{1}{4}
$$

 

Substitute the values into the equation:

$$
1 = \frac{1}{4}(2) + c
$$

$$
1 = \frac{1}{2} + c
$$

$$
c = \frac{1}{2}
$$

 

The equation of the line is

$$
y = \frac{1}{4}x + \frac{1}{2}
$$

 

 

 

Key Points to Remember

Parallel lines have the same gradient.
Perpendicular lines have gradients that multiply to \( -1 \).
To find a perpendicular gradient, use the negative reciprocal.
Once the gradient is known, substitute a point into \( y = mx + c \) to find the equation.

 

Identifying equations of parallel and perpendicular lines depends on understanding gradients and applying them consistently within straight line equations.