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Finding the Slope from a Linear Equation in General Form

(KS3, Year 9)

The slope of a line is its steepness. It is how far up a line goes compared to how far across it goes. The line below has a slope of 2 because it goes up 2 units for every 1 unit it goes across. The slope of the line is 2 because it goes up 2 units for every 1 it goes across

The slope of the line is 2 because it goes up 2 units for every 1 it goes across

A line can be represented by a linear equation. We can find the slope from a linear equation.

How to Find the Slope from a Linear Equation in General Form

A linear equation (in general form) is given in the form below: a x plus b y plus c equals 0

The a, b and c do not immediately tell you anything about the line (such as its slope or y-intercept). But it can be converted to a form which does tell you about the line. For example, the slope-intercept form gives the slope and y-intercept. a x plus b y plus c equals 0 converted to y equals m x plus c

a x plus b y plus c equals 0 converted to y equals m x plus c

It is the m in the slope-intercept form that tells you the slope. Once we know the slope-intercept form, we can read off the number in front of the x (the m) to find the slope. We can use algebra to convert the general form into the slope-intercept form: y equals minus a over b minus c over b

Comparing this with y = mx + c, we see that the slope is −^a⁄_b. Learn more about converting a linear equation in general form to slope-intercept form

A Real Example of Finding the Slope from a Linear Equation in General Form

Finding the slope of a line from a linear equation in general form is easy.

Question

Find the slope of the line given by the linear equation shown below. 4 x plus 2 y plus 1 equals 0

Step-by-Step:

1

Find the number in front of the x. In our example, the number in front of the x is 4.

4x + 2y + 2 = 0

2

Find the number in front of the y. In our example, the number in front of the y is 2.

4x + 2y + 2 = 0

3

Divide the number from Step 1 (4) by the number from Step 2 (2).

4 ÷ 2 = 2

4

Change the sign of the answer from Step 3. In our example, the answer is positive, so change it to negative.

2 → −2

Answer:

The slope is −2.

Lesson Slides

The slider below gives a real example of how to find the slope from a linear equation.

More Examples of Finding the Slope of a Line from Linear Equations

All of the linear equations we have seen in this lesson have been in general form (ax + b + c = 0). There are other forms of linear equation. You must be able to find the y-intercept in all forms of linear equation.