The coordinate plane (also called the xy-plane) has two perpendicular axes:

• x-axis: the horizontal axis
• y-axis: the vertical axis

The point where the two axes intersect is called the origin, with coordinates (0, 0).

A point on the plane is written as (x, y), where:

• x is the horizontal distance from the y-axis (the x-coordinate)
• y is the vertical distance from the x-axis (the y-coordinate)

Example: The point A(2, 3) means x = 2 and y = 3.

The gradient (also called slope) of a line measures how steep the line is. It is the rate of change of y with respect to x.

For two points (x₁, y₁) and (x₂, y₂) on a line, the gradient m is:

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

Example: Find the gradient of the line joining points A(1, 2) and B(4, 8).

$m = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$

Slope-Intercept Form

The most common form is:

$y = mx + c$

Where:

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

Example: In the equation y = 2x + 3, the gradient is 2 and the y-intercept is 3.

Point-Slope Form

When you know the gradient and one point on the line, use:

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

Where (x₁, y₁) is a point on the line and m is the gradient.

You can determine the equation of a straight line if you know any one of these:

1. Gradient and y-intercept
2. Gradient and one point on the line
3. Two points on the line

Case 1: Gradient and y-intercept given

Example: Find the equation of a line with gradient 3 and y-intercept 5.

Since y = mx + c:

• m = 3
• c = 5

Therefore: y = 3x + 5

Case 2: Gradient and a point given

Example: Find the equation of a line with gradient -2 that passes through the point (3, 7).

Using point-slope form: y - y₁ = m(x - x₁)

$y - 7 = -2(x - 3)$

$y - 7 = -2x + 6$

$y = -2x + 13$

Therefore: y = -2x + 13

Case 3: Two points given

Example: Find the equation of a line passing through points (2, 3) and (5, 9).

Step 1: Find the gradient:

$m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2$

Step 2: Use point-slope form with one of the points:

$y - 3 = 2(x - 2)$

$y - 3 = 2x - 4$

$y = 2x - 1$

Therefore: y = 2x - 1

To draw the graph of a straight line:

1. Create a table of values by choosing x-values and calculating corresponding y-values using the equation
2. Plot the points on the coordinate plane
3. Join the points with a straight line

Example: Draw the graph of y = 2x + 1

Step 1: Create a table of values:

Step 2: Plot these points on the graph paper and join them with a straight line.

The line will cross the y-axis at y = 1 (the y-intercept) and have a gradient of 2 (rising 2 units for every 1 unit moved to the right).

• Y-intercept: The point where the line crosses the y-axis (where x = 0)
• X-intercept: The point where the line crosses the x-axis (where y = 0)

Example: Find the intercepts of the line 2x + 3y = 12

For y-intercept (x = 0): $2(0) + 3y = 12$ $3y = 12$ $y = 4$ The y-intercept is 4.

For x-intercept (y = 0): $2x + 3(0) = 12$ $2x = 12$ $x = 6$ The x-intercept is 6.

• The equation of a straight line can be written as y = mx + c (slope-intercept form)
• The gradient m = (y₂ - y₁)/(x₂ - x₁)
• To find the equation, you need either:
  • Gradient and y-intercept
  • Gradient and one point
  • Two points on the line
• To draw the graph, create a table of values, plot the points, and join them with a straight line

In Tanzania, the equation of a straight line is used in many practical situations. For example, if a bus company charges a fixed fare of 2,000 TZS plus 500 TZS per kilometer, the total fare y for a journey of x kilometers can be represented by the equation y = 500x + 2,000. By drawing this graph, the bus company can easily determine the fare for any distance, and passengers can estimate their travel costs before beginning their journey.