### Straight-line Graphs

We all know how straight line graphs span across graph papers - or grids. These graphs have two things in common. They
are **Gradient** and **Intercept.**

#### Gradient - m

The slope or steepness of a
graph is called the gradient. It is calculated by dividing a change in *'y'* by change in *'x'* between two points.

The gradient can be **positive**, **negative** or **zero**.

#### Intercept - c

This is the distance between the origin of a graph and the point where it crosses the y-axis. The intercept can also be negative, positive or zero.

#### The Equation of a Straight Line

**y = mx + c** is the equation of a straight line. Any straight line can be expressed in this form where 'm' and 'c' are gradient and intercept
respectively. The graph takes its shape, depending on the values of the gradient
and the intercept. This following animation shows that.

The gradients of **parallel lines** are the **same**; the **product of the gradients** of **perpendicular lines** is **-1**.

**E.g.1**

The gradient of a straight line is 2. It passes through (3,4). Find its equation.

y = mx + c

4 = 2 X 3 + c

4 = 6 + c

c = -2

The equation is, y = 2x - 2.

**E.g.2**

The gradient of a straight line is -3. It passes through (-2,4). Find its equation.

y = mx + c

4 = -3 X -2 + c

4 = 6 + c

c = -2

The equation is, y = -3x - 2.

**E.g.3**

A straight line passes through (2,5) and (4, 11). Find its equation.

m = (11 - 5) / (4 - 2) = 6 / 2 = 3

y = mx + c

x = 2, y = 5, m = 3,

5 = 3 X 2 + c

5 = 6 + c

c = -1

The equation is y = 3x - 1.

**E.g.4**

The equation of a straight line is y = 3x -2. Find the equation of a second line that passes through (4,-3), parallel to the first line.

Since the two lines are parallel, the gradients are the same. For the second line,

x = 4, y = -3, m = 3

-3 = 3 X 4 + c

-3 = -12 + c

c = 9

The equation is y = 3x + 9.

**E.g.5**

The equation of a straight line is y = 2x -3. Find the equation of a second line that goes through (4,-5), at right angles to the first line.

If the gradient of the second line is m, then 2 X m = -1; => m = -1/2

For the second line,

x = 4, y = -5, m = -1/2

-5 = 4 X (-1/2) + c

-5 = -2 + c

c = -3

The equation is y = -1/2 x - 3.

**E.g.6**

The equation of a straight line is 2y = 3x - 8. Find the equation of a parallel line that goes through (4,3).

Let's rearrange the equation in y = mx + c form.

So, y = 3/2 x - 4
If the gradient of the line is m, then m = 3/2;

For the second, parallel line,

x = 4, y = 3, m = 3/2

3 = 4 X (3/2) + c

3 = 6 + c

c = -3

The equation is y = 3/2 x - 3. or 2y = 3x - 6

**E.g.7**

The equation of a straight line is 3y = 2x -12. Find the equation of a second line that goes through (4,-5), at right angles to the first line.

Let's rearrange the equation in y = mx + c form.

y = 2/3 x - 4

If the gradient of the second line is m, then 2/3 X m = -1; => m = -3/2

For the second line,

x = 4, y = -5, m = -3/2

-5 = 4 X (-3/2) + c

-5 = -6 + c

c = 1

The equation is y = -3/2 x + 1.

**Now, in order to complement what you have just learnt, work out the following questions:**

Find the equations of the following graphs - use the button to get a pair of coordinates.

**Coordinates for a Straight Line Graph**

Use the coordinates to find the equation of the line.