What is the slope formula?
The slope formula is m = (y₂ − y₁)/(x₂ − x₁), the change in y divided by the change in x. It measures how far the line goes up for each step to the right.
Which point you call first does not matter, as long as you use the same order on top and bottom. Swapping both flips both signs, so the slope stays the same.
Try it with (3, 8) first: (2 − 8)/(1 − 3) = −6/−2 = 3. Same answer. The slope of a straight line is the same between any two of its points, which is why two points are enough to pin the line down.
How do you find the equation of a line from two points?
Find the slope, substitute one point into y = mx + b to get b, then write the equation. For (1, 2) and (3, 8) the line is y = 3x − 1.
Check it with the other point: 3(3) − 1 = 8. The line also crosses the x-axis at (1/3, 0) and the y-axis at (0, −1).
What are slope-intercept, point-slope, and standard form?
They are three ways to write the same line. The calculator gives all of them, so you can copy whichever one your teacher asks for.
| Form | General shape | Line through (1, 2) and (3, 8) |
|---|---|---|
| Slope-intercept | y = mx + b | y = 3x − 1 |
| Point-slope | y − y₁ = m(x − x₁) | y − 2 = 3(x − 1) |
| Standard | Ax + By = C | 3x − y = 1 |
Slope-intercept form is best for graphing and reading the y-intercept. Point-slope form is the fastest to write when you know a point and the slope. Standard form keeps x and y together and avoids fractions.
Point-slope form changes depending on which point you use. Through (3, 8) the same line is y − 8 = 3(x − 3). Both versions simplify to y = 3x − 1, so either one is correct.
How do you find the slope from an equation like 2x + 3y = 6?
Solve for y, then read the coefficient of x. For 2x + 3y = 6 the slope is −2/3 and the y-intercept is (0, 2).
Subtract 2x from both sides to get 3y = −2x + 6. Divide every term by 3: y = −(2/3)x + 2. This is what a slope intercept form calculator does when you give it a line in standard form. As a shortcut, the slope of Ax + By = C is always −A/B.
What is the difference between zero slope and undefined slope?
A horizontal line has slope 0, and a vertical line has an undefined slope. The difference comes from which part of the fraction is zero.
For (1, 4) and (5, 4), the rise is 0, so m = 0/4 = 0 and the line is y = 4. For (2, 1) and (2, 7), the run is 2 − 2 = 0. Dividing by zero is undefined, so the line has no slope and its equation is x = 2.
How do you find the y-intercept?
Put x = 0 into the equation, or read b from y = mx + b. For y = 3x − 1, the y-intercept is (0, −1).
If you only have two points, find the slope first and then solve y₁ = m·x₁ + b for b. The calculator works as a y intercept calculator this way, and it shows the substitution step.
For a line in standard form, set x = 0 and solve. In 2x + 3y = 6 that leaves 3y = 6, so y = 2 and the intercept is (0, 2). Setting y = 0 instead gives the x-intercept, (3, 0). Those two points are often the quickest way to graph a line in standard form.
What are common slope mistakes?
Mixing the order is the most common: (8 − 2)/(1 − 3) gives −3 instead of 3. Keep the same point first in both the top and the bottom.
Putting x on top (run over rise) is another. Students also read the slope of 2x + 3y = 6 as 2, forgetting to solve for y first, and call a vertical line's slope 0 when it is undefined.
What does rise over run mean on a graph?
Rise is how far you move up or down between two points, and run is how far you move left or right. Slope is the rise divided by the run.
On the line y = 3x − 1, start at (0, −1). Move 1 unit right and 3 units up, and you land on (1, 2). Do it again and you reach (2, 5). Counting boxes like this on graph paper is a quick way to read a slope without the formula.
A slope of −2/3 works the same way: 3 units right and 2 units down. Keep the sign on the rise and the run positive, and you won't get confused about direction.