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CIE A-Level Maths Study Notes

1.3.1 Equations of a Line

In this section, we explore how to derive the equation of a straight line. Understanding these methods is crucial for solving problems in coordinate geometry. We will cover two primary methods: using two points (via the two-point formula) and using a single point with a known gradient (point-slope form).

The General Form of a Line Equation

  • A line equation is typically expressed as y=mx+cy = mx + c.
  • Here, mm represents the gradient of the line.
  • cc is the y-intercept, the point where the line crosses the y-axis.

Method 1: Using a Point and the Gradient (Point-Slope Form)

When we know a point A(x1,y1)A(x_1, y_1) on the line and its gradient mm, we can express the line's equation as: yy1=m(xx1)y - y_1 = m(x - x_1)

Example 1

Find the equation of the straight line with a gradient of 3 that passes through the point (1, 6).

Solution:

Let A(1,6)A(1, 6) and m=3m = 3.

The line equation becomes:

y6=3(x1)y - 6 = 3(x - 1)

Expanding this, we get:

y=3x3+6y = 3x - 3 + 6

Therefore, the line equation is:

y=3x+3\therefore y = 3x + 3

Method 2: Using Two Points (Two-Point Formula)

When two points, say (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), are given, we first find the gradient mmusing: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} Then, we use this gradient with one of the points in the point-slope form to find the equation.

Example 2

Determine the equation of the line passing through the points (-5, 3) and (-4, 1).

Solution:

First, calculate the gradient mm:

m=134(5)=21=2m = \frac{1 - 3}{-4 - (-5)} = \frac{-2}{1} = -2

The general form of the line is y=2x+cy = -2x + c.

To find cc, substitute one of the points, say (-4, 1):

1=2(4)+c1 = -2(-4) + c

1=8+c1 = 8 + c

c=7 c = -7

Thus, the equation of the line is:

y=2x7 \therefore y = -2x - 7

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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