Linear Equations (in Slope-Point Form)
(KS3, Year 9)

The Lesson

A linear equation is an equation that represents a line. A linear equation can be written in the form: On a graph, a linear equation looks like a line:

• y and x are the Cartesian coordinates of points on the line.
• m is the slope of the line. It tells you the steepness of the line.
• (x₁, y₁) is a point on the line.

A Real Example of a Linear Equation in Slope-Point Form

An example of a linear equation in slope-point form is given below: If we compare this equation to y − y₁ = m(x − x₁), we can find the slope and a point on the line.

• The number in front of the brackets is the slope.
  y − 2 = 2(x − 1)
  The slope is 2. Read more about finding the slope from a linear equation in slope-point form
• A point on the line can be found from the numbers being subtracted from y and x.
  y − 2 = 2(x − 1)
  1 is being subtracted from x. 1 is the x-coordinate of the point. 2 is being subtracted from y. 2 is the y-coordinate of the point. The point on the line is (1, 2). Read more about finding the y-intercept from a linear equation in slope-point form

Other Forms of Linear Equations

There are other forms of linear equation.

• The general form of a linear equation is:
  
  Read more about the general form of a linear equation
• The slope-intercept form of a linear equation is: m is the slope and c is the y-intercept. Read more about the slope-intercept form of a linear equation

Beware

When Points Have Negative Coordinates

In this lesson, we have said that:

• the number that is subtracted from the y gives the y-coordinate of a point.
• the number that is subtracted from the x gives the x-coordinate of a point.

What if a number is added to the y or x? Remember, that subtracting a negative number is the same as adding the positive number: −1 is being subtracted from y, so the y-coordinate is −1. When a number is added to y or x, the coordinate is negative.