Slope-Intercept to Standard Form

Table of Contents

• Slope-intercept form and standard form
• Parts of the calculator
  • Slope (m)
  • Y-intercept (b)
  • A, B and C
  • X-intercept
• How to convert slope-intercept form to standard form
• How to convert standard form to slope-intercept form
• Frequently asked questions

Slope-intercept form and standard form

A straight line can be written in more than one way. The two most common are:

• Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept, the point where the line crosses the y-axis.
• Standard form: Ax + By = C, where A, B and C are whole numbers, A is not negative, and the three have no common factor.

Both describe the same line. Slope-intercept form shows the slope at a glance, which makes it easy to graph; standard form has no fractions, works for vertical lines too, and is the form many algebra classes ask for.

Enter the slope and the y-intercept to get the standard form, or enter A, B and C to get the slope and the y-intercept. To work out the slope and the line from two points, use the slope calculator.

Parts of the calculator

Slope (m)

The slope tells how steep the line is: how much y changes when x grows by 1. A fraction slope can be typed as a decimal with four places: ⅔ as 0.6667, ⅚ as 0.8333.

It can be found with the following formula:

• If you know the standard form:

  m = -A / B

Y-intercept (b)

The y-intercept is the value of y where the line crosses the y-axis, at x = 0.

It can be found with the following formula:

• If you know the standard form:

  b = C / B

A, B and C

The three numbers of the standard form Ax + By = C. The calculator always gives them as whole numbers, with A not negative and no common factor, so 4x + 6y = 12 comes out as 2x + 3y = 6. When you type them yourself, any numbers work, decimals included.

If you know the slope and the y-intercept, the calculator works in the background in four steps:

1. It finds the denominator of m and the denominator of b: the smallest whole number up to 12 that turns each into a whole number (a slope of 0.6667 has a denominator of 3), or the denominator of the exact decimal.

2. Their common denominator, L, is the least common multiple of the two.

3. Moving mx to the left side and multiplying by the common denominator gives A before reducing, B before reducing and C before reducing:

   A = -m * L, B = L, C = b * L

4. Dividing all three by their common factor, and flipping the sign of all three if A is negative, gives the standard form.

X-intercept

The x-intercept is the value of x where the line crosses the x-axis, at y = 0.

It can be found with the following formula:

• If you know the standard form:

  x = C / A

From the slope and the y-intercept it is the same point, x = -b / m; the calculator works it out from the standard form, which has no rounded decimals.

A horizontal line (A = 0) never crosses the x-axis, unless it is the x-axis itself, so it has no x-intercept.

How to convert slope-intercept form to standard form

Example: write y = ⅔x - 4 in standard form.

1. Move the x term to the left. Subtract ⅔x from both sides:

   -⅔x + y = -4

2. Clear the fraction. Multiply every term by the denominator, 3:

   -2x + 3y = -12

3. Make A positive. Multiply every term by -1:

   2x - 3y = 12

The three numbers 2, 3 and 12 have no common factor, so this is the standard form: A = 2, B = -3, C = 12.

How to convert standard form to slope-intercept form

Example: write 4x + 6y = 12 in slope-intercept form.

1. Move the x term to the right. Subtract 4x from both sides:

   6y = -4x + 12

2. Divide by B. Divide every term by 6:

   y = -⅔x + 2

So the slope is m = -⅔ and the y-intercept is b = 2.

If B is 0, as in 3x = 6, the line is vertical, x = 2. A vertical line has no slope and no y-intercept, so it can't be written in slope-intercept form.

Frequently asked questions

What are the rules of standard form?

A, B and C are integers, A is zero or positive, A and B are not both zero, and the three have no common factor other than 1. Some textbooks allow any real numbers; the result here meets the stricter rules.

What if the slope is a fraction?

Multiply the whole equation by the fraction's denominator, as in the example above. When the slope and the intercept have different denominators, multiply by their least common multiple: y = ¼x + ½ becomes x - 4y = -2.

How do you find the slope of a line in standard form?

The slope is m = -A / B. For 2x - 3y = 12, that is -2 / -3 = ⅔.