What slope-intercept form tells you
y = mx + b m is the slope and b is where the line crosses
the y-axis. Those two numbers are everything you need to draw it: start at
(0, b), then move one step right and m units
vertically. That is why this form is useful for graphing.
Finding the equation of a line from two points
This is a common version of the question, and it takes two steps: find the slope, then find where the line crosses the y-axis.
- Slope.
m = (y₂ − y₁) ÷ (x₂ − x₁)— the change in y divided by the change in x. - Intercept.
b = y₁ − m × x₁, using either point. Both give the same answer, which is a useful check.
Example. For (1, 2) and (4, 3): the slope is
(3 − 2) ÷ (4 − 1) = 1/3. Then
b = 2 − (1/3)(1) = 5/3, so the line is
y = (1/3)x + 5/3. Using the second point instead:
b = 3 − (1/3)(4) = 5/3, the same.
The two points tab above does this and shows both steps, along with the same line written in point-slope and standard form.
Two points with the same x-value are the exception: that line is
vertical, has no slope, and cannot be written as y = mx + b at all. Its
equation is x = c where c is the shared x-value.
Rearranging an equation into this form
The work is always the same: collect the x and y terms, get the y term alone on one side, then divide by its coefficient.
For 2x + 3y = 6: subtract 2x from both sides to
get 3y = −2x + 6, then divide everything by 3, giving
y = (−2/3)x + 2. The slope is −2/3 and the line
crosses the y-axis at 2.
The step people most often miss is dividing every term by the
coefficient, not just the x term. Dividing 3y = −2x + 6
by 3 has to change the 6 as well.
What this calculator will read
- Standard form —
2x + 3y = 6,-2x + 5y = 10 - Point-slope form —
y - 3 = 2(x - 1) - Already solved —
y = 4x - 5, which it will simply confirm - Terms on both sides —
3x + 2 = y - x - Fractions and decimals —
2/3x + y = 4,0.5x + 1.5y = 3 - Vertical lines —
3x = 9, which it will tell you has no slope-intercept form
The cases that aren’t lines
Some equations look linear and describe no line at all, and a calculator
that returns m = 0, b = 0 for them is hiding the answer
rather than giving it.
- Vertical lines such as
3x = 9have no slope, so noy = mx + bexists. The equation isx = 3and nothing else. - Identities such as
x + y = x + yare true for every point on the plane. That is not a line. - Contradictions such as
x + 1 = x + 5are true for no point at all, so there is nothing to draw.
Why fractions instead of decimals
The slope of 2x + 3y = 6 is exactly −2/3. Rounded to
−0.67 it is a different number, and the error carries into anything
you calculate afterwards. This calculator keeps calculated values exact,
displaying fractions instead of rounding them to decimals.
Frequently asked questions
How do I convert standard form to slope-intercept form?
Move the x term to the other side, then divide every term by the coefficient of y. For Ax + By = C the result is y = (−A/B)x + C/B. Dividing every term — including the constant — is the step most often missed.
How do I find the equation of a line with two points?
Find the slope first with m = (y₂ − y₁) ÷ (x₂ − x₁), then put either point into b = y₁ − m × x₁ to get the y-intercept. The equation is y = mx + b. Both points give the same intercept, so using the other one checks your work.
What do m and b mean?
m is the slope — how far the line rises for each step to the right. b is the y-intercept, the y-value where the line crosses the y-axis, which is the point (0, b).
Can every line be written as y = mx + b?
No. Vertical lines cannot, because they have no slope — one x-value corresponds to every y, so y is not a function of x. Those lines are written x = c instead.
What is b when the line passes through the origin?
Zero, and the equation shortens to y = mx. A line through the origin crosses the y-axis at (0, 0), so there is no constant to add.