Slope and the equation of a line

Slope is how much y changes for each step you take to the right in x. People often say it as “rise over run”: how far the line goes up or down, divided by how far it goes across. The letter for slope is m.

A slope of 2 means the line goes up 2 for every 1 step right. A slope of −1/3 means it goes down 1 for every 3 steps right.

Given two points (x₁, y₁) and (x₂, y₂), the slope is the change in y divided by the change in x: m = (y₂ − y₁)/(x₂ − x₁). It does not matter which point you call first, as long as you use the same order on the top and the bottom.

• Subtracting a negative is adding: 4 − (−2) = 6. This is the most common slip.
• Keep the y values on top. Run over rise gives the wrong answer.

The most common way to write a line is y = mx + b. Here m is the slope and b is the y-intercept, the y value where the line crosses the y axis. This form is handy because you can read both numbers straight off the equation and graph the line right away.

If a line comes in another form, like 2x + 3y = 6, solve for y to get it into slope-intercept form: 3y = −2x + 6, so y = −(2/3)x + 2. The slope is −2/3 and the line crosses the y axis at 2.

Find the slope first, then use one point to find b. Put the point’s x and y into y = mx + b and solve for b.

Point-slope form is y − y₁ = m(x − x₁), where (x₁, y₁) is any point on the line. It is the fastest form to write when you know a point and the slope, because you just fill in the numbers. You can then simplify it into y = mx + b if your teacher wants that form.

1. Find where the line crosses the y axis. That y value is b.
2. From that point, find another point where the line crosses the grid exactly at a corner.
3. Count the rise, up is positive and down is negative, and the run to the right.
4. Slope = rise ÷ run.
5. Write y = mx + b and check that another grid point fits.

A horizontal line has the same y value everywhere, so its slope is 0 and its equation is just y = a number, like y = 3. A vertical line has the same x value everywhere. Its run is always 0, and you cannot divide by 0, so its slope is undefined. Its equation is x = a number, like x = −4.

An easy way to keep them apart: y = 3 is the set of all points whose y is 3, which lie in a flat row. x = −4 is all points whose x is −4, which lie in a standing column.

Parallel lines never meet, so they have the same slope. Perpendicular lines cross at a right angle, and their slopes are negative reciprocals: flip the fraction and change the sign. Two perpendicular slopes always multiply to −1, unless one line is horizontal and the other vertical.