How to Convert a Linear Equation from General to Slope-Intercept Form

Converting a Linear Equation from General to Slope-Intercept Form

A linear equation is an equation that represents a line.

We can convert a linear equation from general form to slope-intercept form.

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

Why Convert from General Form to Slope-Intercept Form?

The same line can be written in general form and slope-intercept form.

The two forms of linear equation tell us different things about the line.

  • In general form, a, b and c do not tell us anything about the line.

    a x plus b y plus c equals 0 on a graph
  • In slope-intercept form,

    m tells us the slope of a line

    c tells us the y-intercept of a line.

    y equals m x plus c on a graph

If we are given a line in general form, we can convert it to slope-intercept form to understand more about the line.

How to Convert a Linear Equation from General to Slope-Intercept Form

Using algebra, we can take the general form of a linear equation and make it look like the slope intercept form.

  • Start with the linear equation in general form:

    a x plus b y plus c equals 0
  • Subtract ax from both sides. This has the effect of moving the ax to the other side of the equals sign (=) and changing its sign from positive to negative:

    b y plus c equals minus a x
  • Subtract c from both sides:

    b y equals minus a x minus c
  • Divide both sides by b:

    y equals minus a over b x minus c over b

We can compare this equation with the slope-intercept form of a linear equation:

Comparing a x plus b y plus c equals zero with y equals m x plus c equals zero
  • The slope is given by ab.

  • The y-intercept is given by cb.

A Real Example of How to Convert a Linear Equation from General to Slope-Intercept Form

Convert the linear equation in general form below to slope-intercept form.

Question

Find the y-intercept of the line given by the linear equation shown below.

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

Step-by-Step:

1

Subtract 4x from both sides:

4x + 2y + 6 − 4x = 0 − 4x

2y + 6 = −4x

2

Subtract 6 from both sides:

2y + 6 − 6 = − 4x − 6

2y = −4x − 6

3

Divide both sides by 2:

2y ÷ 2 = (− 4x − 6) ÷ 2

y = −2x − 3

Answer:

We have converted 4x + 2y + 6 (in general form) to y = −2x − 3 (in slope-intercept form).

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See Also

What is a linear equation? What is a linear equation in general form? What is a linear equation in slope-intercept form?