Standard Form to Slope-Intercept Form Calculator
Enter the coefficients A, B, and C from the standard form equation Ax + By + C = 0. The calculator converts the equation to slope-intercept form y = mx + b, isolating the slope and y-intercept, and also gives you the x-intercept and the angle the line makes with the horizontal. A step-by-step panel shows every algebraic move, and a live graph displays the resulting line.
What is standard form and slope-intercept form?
A linear equation can be written in several equivalent ways. Standard form is Ax + By + C = 0, where A, B, and C are real numbers and A and B are not both zero. It is compact and works for every line, including vertical lines (when B = 0). Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. This form is the most widely used in algebra classes because slope and intercept are visible at a glance without any further calculation. Both forms represent the same straight line; they are just different ways of writing the same relationship between x and y.
How to convert standard form to slope-intercept form
To convert Ax + By + C = 0 into y = mx + b, isolate y. Start by moving the x-term and constant to the right side: By = -Ax - C. Then divide every term by B: y = (-A/B)x + (-C/B). The slope is m = -A/B and the y-intercept is b = -C/B. For example, given 2x + 3y - 6 = 0, divide through by 3 to get y = (-2/3)x + 2. The slope is -2/3 and the y-intercept is 2. Note that this conversion requires B to be non-zero. When B = 0 the equation describes a vertical line, which has no defined slope and cannot be written in slope-intercept form.
How to convert slope-intercept form to standard form
To convert y = mx + b back into standard form, move all terms to the left side: mx - y + b = 0. This gives A = m, B = -1, and C = b. Some textbooks prefer integer coefficients: if m is a fraction p/q, multiply the whole equation by q. For example, y = (2/3)x + 4 becomes 3y = 2x + 12, then 2x - 3y + 12 = 0. The conversion is always possible, and the resulting standard form describes exactly the same line.
Slope, intercepts, and the angle of the line
Once you have slope-intercept form, three additional properties follow directly. The x-intercept, where the line crosses the x-axis, is found by setting y = 0 and solving: x = -b/m (undefined when m = 0, which means the line is horizontal and never crosses the x-axis unless b is also 0). The percent slope expresses the gradient as a percentage - a slope of 0.5 is a 50 percent grade. The angle the line makes with the positive x-axis is the inverse tangent of the slope: theta = arctan(m), converted to degrees by multiplying by 180 divided by pi. A slope of 1 gives a 45-degree angle; a slope of 0 gives a horizontal line at 0 degrees.
Relationship between the two linear equation forms
| Property | Standard form: Ax + By + C = 0 | Slope-intercept form: y = mx + b |
|---|---|---|
| Slope | -A / B | m (direct) |
| Y-intercept | -C / B | b (direct) |
| X-intercept | -C / A | -b / m |
| Vertical line | B = 0 (e.g., x = 3) | Not expressible |
| Horizontal line | A = 0 (e.g., y = 5) | m = 0 (e.g., y = 5) |
| Conversion to SI | y = (-A/B)x + (-C/B) | Already in y = mx + b |
| Conversion to standard | Already in Ax + By + C = 0 | mx - y + b = 0 |
Both forms represent the same straight line. Standard form uses integer coefficients; slope-intercept form makes slope and intercept immediately readable.
Frequently asked questions
What is the formula to convert standard form to slope-intercept form?
Starting from Ax + By + C = 0, isolate y: By = -Ax - C, then divide by B: y = (-A/B)x + (-C/B). The slope is m = -A/B and the y-intercept is b = -C/B. This works for any equation where B is not zero.
What happens when B equals zero in standard form?
When B = 0 the equation Ax + C = 0 describes a vertical line at x = -C/A. Vertical lines have an undefined slope, so they cannot be written in slope-intercept form y = mx + b. This calculator flags that case and explains why the conversion is not possible.
Can I convert slope-intercept form back to standard form?
Yes. Starting from y = mx + b, move all terms to the left: mx - y + b = 0. This is already standard form with A = m, B = -1, and C = b. If you prefer integer coefficients and m is a fraction, multiply every term by the denominator of m to clear it.
How do I find the x-intercept from either form?
Set y = 0 and solve for x. In slope-intercept form: 0 = mx + b, so x = -b/m. In standard form: Ax + C = 0, so x = -C/A. The result is the same from both forms because they represent the same line.
What does the angle output represent?
The angle is the inclination of the line - how many degrees it tilts from the horizontal (positive x-axis direction). It is calculated as arctan(m) and converted to degrees. A slope of 0 gives 0 degrees (flat line), a slope of 1 gives 45 degrees, and a very large positive slope gives an angle approaching 90 degrees.
Why does slope-intercept form use 'y = mx + b' and not another letter?
The letters m for slope and b for intercept are conventions in American and British algebra textbooks. In other countries or textbooks you may see y = kx + d, y = ax + c, or similar. The variable names do not affect the math - the structure y = (slope)x + (y-intercept) is what matters.