Equations of Linear Graphs

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Understanding the Equation y = mx + c

Every straight line on a graph has an equation that describes it. The most common form of this equation is:

y = mx + c

In this equation:

• m represents the gradient (the steepness or slope of the line)
• c represents the y-intercept (where the line crosses the y-axis)

Think of it like this: the gradient tells you how steep the line is, and the y-intercept tells you where the line starts on the vertical axis.

Examples:

• The line y = 3x + 2 has a gradient of 3 and crosses the y-axis at 2
• The line y = -5x + 7 has a gradient of -5 and crosses the y-axis at 7
• The line y = x - 4 has a gradient of 1 (because x means 1x) and crosses the y-axis at -4

What is the Gradient (m)?

The gradient measures how steep a line is. It tells you how much the line goes up (or down) for every step you take to the right.

How to calculate gradient:

Gradient = change in y ÷ change in x

Or you can remember it as: rise ÷ run

• The rise is how much you go up or down (vertical distance)
• The run is how much you go across (horizontal distance)

Important rules about gradient:

• A positive gradient means the line slopes upward (from bottom-left to top-right)
• A negative gradient means the line slopes downward (from top-left to bottom-right)
• The bigger the number, the steeper the line
• A gradient of 2 is steeper than a gradient of 1
• A gradient of -5 is steeper than a gradient of -3

Formula for gradient between two points:

If you know two points on a line, such as (x₁, y₁) and (x₂, y₂), you can find the gradient using:

m = (y₂ - y₁) ÷ (x₂ - x₁)

Example: Find the gradient between points (2, 5) and (6, 13).

m = (13 - 5) ÷ (6 - 2) = 8 ÷ 4 = 2

The gradient is 2.

Finding the Equation from a Graph

When you're given a graph of a straight line, you can find its equation by following these steps:

Step 1: Find the gradient (m)

• Choose any two clear points on the line
• Draw a right-angled triangle between these points
• Count the vertical distance (rise) and horizontal distance (run)
• Calculate: gradient = rise ÷ run
• Check if the line goes upward (positive) or downward (negative)

Step 2: Find the y-intercept (c)

• Look at where the line crosses the y-axis
• Read off the y-value at this point

Step 3: Write the equation

• Substitute your values of m and c into y = mx + c

Example: A line passes through (0, 3) and (2, 7).

• Rise = 7 - 3 = 4
• Run = 2 - 0 = 2
• Gradient = 4 ÷ 2 = 2
• The line goes upward, so m = 2
• The line crosses the y-axis at 3, so c = 3
• Equation: y = 2x + 3

What if the y-intercept is not visible on the graph?

Sometimes the graph doesn't show where the line crosses the y-axis. In this case:

Step 1: Find the gradient as normal

Step 2: Use any point on the line and substitute it into y = mx + c

Step 3: Solve the equation to find c

Example: A line has gradient 4 and passes through the point (3, 15). Find its equation.

• We know m = 4, so we can write: y = 4x + c
• Substitute x = 3 and y = 15: 15 = 4(3) + c
• Simplify: 15 = 12 + c
• Solve for c: c = 15 - 12 = 3
• Equation: y = 4x + 3

Finding the Gradient and y-intercept from an Equation

If you're given an equation, you can identify the gradient and y-intercept by comparing it to y = mx + c.

For equations already in the form y = mx + c:

Simply read off the values:

• The number in front of x is the gradient (m)
• The constant term is the y-intercept (c)

Examples:

• y = 6x + 3: gradient = 6, y-intercept = 3
• y = -2x + 5: gradient = -2, y-intercept = 5
• y = ½x - 1: gradient = ½, y-intercept = -1
• y = 3 - 4x: This can be rewritten as y = -4x + 3, so gradient = -4, y-intercept = 3

Horizontal and Vertical Lines

Horizontal lines are flat lines that go straight across. They have:

• A gradient of 0 (no slope at all)
• An equation in the form y = c

Example: The line y = 5 is a horizontal line that crosses the y-axis at 5. Every point on this line has a y-coordinate of 5, such as (1, 5), (2, 5), (100, 5).

Vertical lines are lines that go straight up and down. They have:

• An undefined gradient (you can't calculate it because you'd be dividing by zero)
• An equation in the form x = k

Example: The line x = -2 is a vertical line that crosses the x-axis at -2. Every point on this line has an x-coordinate of -2, such as (-2, 0), (-2, 5), (-2, -10).

**(Extended) Converting to y = mx + c Form

To find the gradient and y-intercept from an equation in general form, you need to rearrange it to make y the subject.

Steps:

1. Move all terms with x to the other side
2. Divide everything by the number in front of y
3. Simplify to get y = mx + c
4. Read off the gradient and y-intercept

Example 1: Find the gradient and y-intercept of 5x + 4y = 8

• Move 5x to the other side: 4y = -5x + 8
• Divide everything by 4: y = -5x/4 + 8/4
• Simplify: y = -⁵⁄₄x + 2
• Gradient = -⁵⁄₄, y-intercept = 2

Example 2: Find the gradient and y-intercept of 3x - 2y = 12

• Move 3x to the other side: -2y = -3x + 12
• Divide everything by -2: y = (-3x + 12) ÷ (-2)
• Simplify: y = ³⁄₂x - 6
• Gradient = ³⁄₂, y-intercept = -6