The Straight Line on Coordinate Geometry – SS1 Further Mathematics Lesson Note (Second Term)

A straight line on a coordinate plane has a gradient, which is the rate of change of y compared with x, and a y-intercept, which is the value of y where the line crosses the y-axis. Its equation is y = mx + c, where m is the gradient and c is the y-intercept.

Behavioural objectives

By the end of the lesson, students should be able to:

1. find the gradient of a line joining two points;
2. find the gradient and y-intercept of a line from its equation;
3. sketch the graph of a straight line;
4. write the equation of a line parallel to either axis;
5. find the equation of a line from its gradient and one point on it;
6. find the angle of slope of a line.

What are the key terms and formulae for a straight line?

The key terms are gradient, intercept and angle of slope, and each has a standard formula.

• Gradient (m): the rate of change of y compared with x. It is also called the slope.
• y-intercept (c): the value of y where the line crosses the y-axis, that is, where x = 0.
• x-intercept: the value of x where the line crosses the x-axis, that is, where y = 0.
• Angle of slope (θ): the angle a line makes with the positive direction of the x-axis.

What is the gradient of a straight line?

The gradient of a straight line is the rate of change of y compared with x.

• Meaning: if the gradient is 2, then for any increase in x, y increases two times as much.
• Rule: gradient = (increase in y from A to B) / (increase in x from A to B).
• With coordinates: for A(x₁, y₁) and B(x₂, y₂), m = (y₂ – y₁) / (x₂ – x₁).

What are the kinds of gradient?

A line has a positive, negative, zero or undefined gradient, depending on its direction.

Example 1: Gradient from two points

Find the gradient of the line joining P(7, -2) and Q(-1, 2).

1. Let (x₁, y₁) = (7, -2) and (x₂, y₂) = (-1, 2).
2. Increase in y = 2 – (-2) = 4.
3. Increase in x = -1 – 7 = -8.
4. Gradient of PQ = 4 / (-8) = -1/2.

Example 2: Gradient and y-intercept from an equation

Find the gradient of the line 7x + 4y – 8 = 0.

1. Make y the subject: 4y = -7x + 8.
2. Divide by 4: y = -7/4 x + 2.
3. Compare with y = mx + c.
4. Therefore, gradient (m) = -7/4 and y-intercept (c) = 2.

How do you sketch the graph of a straight line?

To sketch a straight line, find two points on it, plot them and join them with a ruler.

• From y = mx + c: for y = 3x – 2, the gradient is 3 and the y-intercept (c) is -2.
• From a general equation: for 2x + 3y = 6, rearrange to y = -2/3 x + 2. The gradient is -2/3 and the y-intercept (c) is 2.
• Intercept method: put x = 0 to find the y-intercept, then put y = 0 to find the x-intercept.

Example: Sketch 4x – 3y = 12

1. When x = 0, -3y = 12, so y = -4. The line crosses the y-axis at (0, -4).
2. When y = 0, 4x = 12, so x = 3. The line crosses the x-axis at (3, 0).
3. Plot (0, -4) and (3, 0), then join them with a straight line.
4. Gradient m = (y₂ – y₁) / (x₂ – x₁) = (0 + 4) / (3 – 0) = 4/3.
5. The y-intercept is -4.

What are lines parallel to the axes?

A line parallel to the x-axis has the equation y = c, and a line parallel to the y-axis has the equation x = a.

Line parallel to the x-axis

• Gradient: zero.
• Equation: y = c, where c may be any number.
• y = 5: a horizontal line through (0, 5).
• y = -3: a horizontal line through (0, -3).

Line parallel to the y-axis

• Gradient: undefined.
• Equation: x = a, where a may be any number.
• x = 2: a vertical line through (2, 0).
• x = -4: a vertical line through (-4, 0).
• The y-axis: its equation is x = 0.
• The x-axis: its equation is y = 0.

What is the equation of a straight line?

The equation of a straight line has the form y = mx + c, where m is the gradient and c is the y-intercept.

How do you find the gradient and y-intercept from an equation?

Compare the given equation with y = mx + c and read off m and c.

Find the gradient and the intercept on the y-axis of each line:

1. y = 3x – 4
2. y = -1/2 x – 3

• y = 3x – 4: the gradient is 3 and the intercept on the y-axis is -4.
• y = -1/2 x – 3: the gradient is -1/2 and the intercept on the y-axis is -3.

Example: Equation from a gradient and a point

Determine the equation of a straight line whose gradient is -1/3 and which passes through the point (-3, 2).

1. Use y – y₁ = m(x – x₁), where (x₁, y₁) = (-3, 2) and m = -1/3.
2. y – 2 = -1/3 (x + 3).
3. Multiply both sides by 3: 3y – 6 = -x – 3.
4. Collect terms: x + 3y = 3.

How do you find the equation of a line from its gradient and one point?

Substitute the gradient and the coordinates of the point into y – y₁ = m(x – x₁), then simplify.

Find the equation of a straight line of slope 2 that passes through the point (3, -2).

1. Write m = 2, x₁ = 3 and y₁ = -2.
2. y – (-2) = 2(x – 3).
3. y + 2 = 2x – 6.
4. y = 2x – 6 – 2.
5. Hence the equation is y = 2x – 8.

How do you find the angle of slope of a line?

The angle of slope θ of a line is found from tan θ = m, where m is the gradient.

• Positive gradient: θ is an acute angle.
• Negative gradient: θ is measured anticlockwise from the positive x-axis, so it is obtuse.

Example

Find the gradient of the line joining (3, 2) and (7, 10) and the angle of slope of the line.

1. Let m be the gradient. Then m = (10 – 2) / (7 – 3).
2. m = 8/4 = 2.
3. Let θ be the angle of slope. Then tan θ = 2.
4. θ = 63.43° (to two decimal places).

Evaluation

1. State the gradient of a line parallel to the x-axis and the form of its equation.
2. Find the gradient and y-intercept of the line 3x + 2y – 6 = 0.
3. Find the equation of the line with gradient 2 that passes through (1, 4).
4. Find the gradient of the line joining (2, 1) and (4, 7), and the angle of slope of the line.

Assignment

Do all three parts in your exercise book.

1. Sketch the graph of the line 3x – 2y = 6.
2. Find the equation of the line with gradient -2 that passes through (-1, 3).
3. Draw the graphs of y = 2 and x = -3 on the same axes.