Slope-Intercept and Standard Form Converter

How to use the slope-intercept and standard form converter

• To go from slope-intercept to standard form: fill in m (slope) and b (y-intercept), and leave A, B, and C blank.
• To go from standard form to slope-intercept: fill in A, B, and C, and leave m and b blank.
• Do not enter both sets of values at the same time. Supplying conflicting inputs or leaving only a single random value filled in will trigger an error message.
• The tool keeps fractions exact and simplifies integer coefficients so they share no common factor.
• When the line is vertical (B = 0 in standard form), there is no valid slope-intercept form. The calculator detects this and reports that the slope is undefined.

Slope-intercept form

Slope-intercept form writes a line as

y = mx + b

• m is the slope: the change in y for each 1-unit change in x.
• b is the y-intercept: the value of y when x = 0.

This form is especially useful when you want to graph quickly or interpret the rate of change and starting value in a word problem.

Standard form

Standard form writes the same line as

Ax + By = C

• A, B, and C are real numbers, often chosen as integers.
• Many textbooks prefer A, B, and C to be integers with no common factor and A > 0.
• When B = 0, the line is vertical (x = C / A) and has undefined slope.

Standard form is convenient when you solve systems of equations by elimination or when you model constraints using integer coefficients.

Formulas for converting between forms

From slope-intercept to standard form

Start with the slope-intercept equation:

y = mx + b

Rearrange to move the x-term to the left:

-mx + y = b

To match Ax + By = C, you can multiply the entire equation by -1 so that the coefficient of x is positive:

mx - y = -b

In general, if you begin with y = mx + b, an equivalent standard form is

mx - y = -b

When m or b are fractions, the calculator multiplies through by a common denominator to clear fractions and then divides by the greatest common divisor so that A, B, and C are simplified integers.

From standard form to slope-intercept

Start with the standard equation:

Ax + By = C

Subtract Ax from both sides:

By = C - Ax

Now divide every term by B (assuming B \neq 0):

y = -A/B x + C/B

So the slope-intercept parameters are

• slope: m = -A/B
• intercept: b = C/B

In MathML, this relationship can be written as:

If B = 0, the line is vertical, has an undefined slope, and cannot be written in slope-intercept form. The calculator reports this case explicitly.

Interpreting the results

The converter does a few things automatically so your answers are clean and consistent:

• Exact arithmetic: it keeps fractions instead of rounding to decimals, so you do not lose precision.
• Simplified standard form: it uses the greatest common divisor (GCD) to remove any common factor shared by A, B, and C.
• Sign convention: if necessary, it multiplies the equation by -1 so that A > 0, which is the most common classroom convention.
• Clear error messages: if the inputs are incomplete, conflicting, or lead to a vertical line when you request a slope, the tool explains what went wrong.

When you see the output, you can treat it as a canonical version of the same line: different algebraic forms, but the same graph.

Worked examples

Example 1: Slope-intercept to standard form

Suppose you have the line

y = (3/4)x - 2

1. Move the x-term to the left:
   - (3/4)x + y = -2
2. Clear the fraction by multiplying every term by 4:
   -3x + 4y = -8
3. Make A positive by multiplying by -1:
   3x - 4y = 8

The standard form is 3x - 4y = 8 with A = 3, B = -4, C = 8. The calculator performs these steps automatically and also checks whether 3, -4, and 8 share any common factor (they do not).

Example 2: Standard form to slope-intercept

Now start from a standard form equation:

2x + 3y = 12

1. Subtract 2x from both sides:
   3y = 12 - 2x
2. Divide by 3:
   y = 12/3 - (2/3)x
3. Simplify:
   y = 4 - (2/3)x

Reordering terms gives

y = -(2/3)x + 4

So the slope-intercept parameters are m = -2/3 and b = 4. When you enter A = 2, B = 3, and C = 12 in the calculator, it will return exactly these values.

Example 3: Vertical line limitation

Consider the equation

5x = 10

This can be written as standard form with A = 5, B = 0, and C = 10. Dividing by 5 gives x = 2, which is a vertical line. There is no way to write this as y = mx + b because the slope is undefined. If you enter A = 5, B = 0, and C = 10, the converter will tell you that the slope-intercept form does not exist for this line.

Comparison of slope-intercept and standard form

Limitations and assumptions

• Input completeness: you must provide either both m and b, or all of A, B, and C. Any other combination is treated as invalid.
• Vertical lines: when B = 0 in standard form, the line is vertical. The calculator cannot convert such a line into slope-intercept form and instead reports that the slope is undefined.
• Standard form convention: the tool aims for integer coefficients with no common factor and A > 0. If your input already satisfies these rules, the output may look the same as your input.
• No rounding by default: the calculator keeps exact rational values wherever possible. If you need decimals, you can convert the fractions on your own according to your required precision.
• Linear equations only: the formulas assume a linear relationship in two variables. Nonlinear equations (quadratic, absolute value, etc.) are outside the scope of this tool.

These assumptions match the way most algebra courses define and use slope-intercept and standard form, so you can rely on the outputs for homework, teaching, and quick checks.