Explore the basic tenets of coordinate geometry (gradient and equations of a straight line, graphs of linear equations)

The coordinate plane is also called the xy-plane. It has two perpendicular lines (axes):

• The x-axis runs horizontally (left to right)
• The y-axis runs vertically (up and down)

The point where these axes cross is called the origin, written as (0, 0).

Every point on the plane is identified by an ordered pair (x, y), where:

• The first number (x) tells you how far to move left or right from the origin
• The second number (y) tells you how far to move up or down

For example:

• Point A(2, 3) means move 2 units right and 3 units up
• Point B(-3, -1) means move 3 units left and 1 unit down

The gradient (also called slope) of a line measures how steep the line is. It tells us how much y changes when x changes by one unit.

Gradient Formula

For two points (x₁, y₁) and (x₂, y₂) on a line:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

where m represents the gradient.

Worked Example

Find the gradient of the line joining points A(2, 3) and B(4, 4).

$m = \frac{4 - 3}{4 - 2} = \frac{1}{2}$

The gradient is ½, meaning for every 2 units we move right, the line goes up by 1 unit.

Types of Gradient

• Positive gradient: Line goes upward from left to right
• Negative gradient: Line goes downward from left to right
• Zero gradient: Horizontal line (y stays the same)
• Undefined gradient: Vertical line (x stays the same)

A straight line can be written in different forms:

Slope-Intercept Form

$y = mx + c$

• m = gradient (slope)
• c = y-intercept (where the line crosses the y-axis)

For example, y = 2x + 3 has a gradient of 2 and crosses the y-axis at 3.

Point-Slope Form

When we know the gradient and one point on the line:

$y - y_1 = m(x - x_1)$

where (x₁, y₁) is a point on the line.

Worked Example

Find the equation of a line with gradient -1 passing through point (4, 2).

Using the point-slope form:

$y - 2 = -1(x - 4)$ $y - 2 = -x + 4$ $y = -x + 6$

Finding Equation from Two Points

Find the equation of the line passing through (1, 2) and (3, 6).

Step 1: Find the gradient

$m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2$

Step 2: Use point-slope form with one point

$y - 2 = 2(x - 1)$ $y - 2 = 2x - 2$ $y = 2x$

• x-intercept: Where the line crosses the x-axis (where y = 0)
• y-intercept: Where the line crosses the y-axis (where x = 0)

Worked Example

Find the y-intercept of the line 2x - 3y = 6

Set x = 0:

$2(0) - 3y = 6$ $-3y = 6$ $y = -2$

The y-intercept is -2, so the line crosses the y-axis at the point (0, -2).

To draw the graph of a linear equation (like y = 2x + 1):

1. Find the y-intercept (c = 1): Plot the point (0, 1)
2. Use the gradient (m = 2 = ²/₁): From (0, 1), move right 1 unit and up 2 units to get another point (1, 3)
3. Draw the line connecting these points

• Parallel lines: Never meet and have the same gradient
• Perpendicular lines: Meet at a right angle; their gradients multiply to give -1

If one line has gradient m, a perpendicular line has gradient -1/m.

Coordinate geometry is used in everyday life in Tanzania. For example, when a bus driver follows a route on a map, the locations of bus stops can be described using coordinates. Similarly, surveyors in Tanzania use coordinate geometry to measure and map land boundaries for building houses or roads. A mason might use the gradient concept to ensure a roof has the correct slope for water to flow off during the rainy season.