Slope, intercepts and parallel lines
A line question is rarely about the line. It is about one number inside it: the slope, an intercept, or the slope of a second line that has to behave a certain way. Knowing which number the question wants is most of the work.
What slope measures
Slope is the change in y for each one unit of change in x. Written as a fraction it is rise over run, and from two points it is (y₂ − y₁) / (x₂ − x₁). A positive slope rises left to right, a negative slope falls, a slope of zero is horizontal, and a vertical line has no slope at all. Keep the points in the same order in the numerator and the denominator, because swapping one of them flips the sign.
Two forms, two different answers
Slope intercept form, y = mx + b, hands you the slope and the y intercept directly, which is why it is the form to reach for when those are the answers. Standard form, ax + by = c, is faster for intercepts: set x to zero to get the y intercept, and set y to zero to get the x intercept. Converting between the two is one or two lines of algebra, so the cost of choosing is small, but choosing badly costs a minute.
Parallel and perpendicular
Parallel lines share a slope. Perpendicular lines have slopes that multiply to negative one, so the slope you want is the negative reciprocal. A line perpendicular to one with slope 3 has slope -1/3. A horizontal line is perpendicular to a vertical one, which is the one case where the reciprocal rule has no number to use.
- Parallel to y = 3x + 2: slope 3
- Perpendicular to y = 3x + 2: slope -1/3
- Perpendicular to y = -x + 4: slope 1
- Horizontal line: slope 0. Vertical line: undefined
Slope with units, and the intercept in context
In a word problem, the slope is a rate and it carries units: dollars per hour, meters per second, points per game. The intercept is the value when the input is zero, which usually reads as a starting amount or a fixed fee. A negative slope means the quantity falls as the input rises, so a draining tank has a positive height and a negative slope. Both facts are usually asked directly, in words rather than symbols.
The fastest route to an equation
When you have a point and a slope, write y − y₁ = m(x − x₁) and stop there. Expanding it into slope intercept form is only worth the algebra if the question asks for the intercept. When you have two points, get the slope first, then reuse either point in the same equation.
How do I find the slope from two points?
Subtract the y values and divide by the difference of the x values, keeping the points in the same order: (y₂ − y₁) / (x₂ − x₁).
What slope is perpendicular to y = 3x + 1?
Negative one third. Flip the fraction and change its sign, so 3 becomes -1/3.
How do I find an intercept quickly?
Set the other variable to zero. In standard form, setting x to zero gives the y intercept in one step, and setting y to zero gives the x intercept.
What does the slope mean in a word problem?
The change in the output for each one unit increase in the input, in the units the problem uses. If the model charges 22 dollars per month, the slope is 22 dollars per month, not 22.
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