Lines & slope
Slope, intercepts, and perpendicular lines.
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the idea
A straight line has one steady steepness, its slope: the change in y divided by the change in x, or rise over run.
more of the idea
Most lines can be written y = mx + b, where m is the slope and b is the y-intercept, the height where the line crosses the y-axis. The x-intercept is where it crosses the x-axis, so y is 0 there.
Perpendicular lines, which meet at a right angle, have slopes that are negative reciprocals of each other.
techniques
Rise over run
- Rise: the second y-value minus the first.
- Run: the second x-value minus the first, in the same order.
- Divide the rise by the run, keep the sign, and reduce the fraction.
What is the slope of the line through (1, −2) and (4, 7)?
- Rise: 7 − (−2) = 9.
- Run: 4 − 1 = 3.
- Slope: 9 ÷ 3 = 3.
Answer: 3
Put the point in
- For b: write y = mx + b with the slope in place of m.
- Put in the point, x and y, and solve for b.
- For the x-intercept: set y to 0 and solve for x. Leave a fraction in lowest terms.
A line has slope 2 and passes through (3, 1). What is its y-intercept?
- Write y = 2x + b and put in the point: 1 = 2 × 3 + b.
- So 1 = 6 + b, and b = 1 − 6 = −5.
Answer: −5
Flip and change the sign
- Flip the slope upside down: 2/3 becomes 3/2, and 4 becomes 1/4.
- Change its sign: positive becomes negative, negative becomes positive.
- Check: the two slopes multiply to −1.
A line has slope −4/3. What is the slope of a line perpendicular to it?
- Flip −4/3 to −3/4.
- Change the sign: 3/4.
- Check: −4/3 × 3/4 = −1.
Answer: 3/4
watch out for
- Dividing the run by the rise. Slope puts the change in y on top.
- Subtracting in one order on top and the other order on the bottom. Start from the same point in both.
- Flipping a slope for a perpendicular line but keeping its sign.
- Giving the y-intercept when the question asks where the line crosses the x-axis.
practice
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Slope from two points
worked example
What is the slope of the line through (2, −8) and (5, 5)? Give a fraction or a whole number.
- Rise over run: (5 − (−8)) ÷ (5 − 2) = 13 ÷ 3 = 13/3.
Answer: 13/3
y-intercept
worked example
A line has slope 4 and passes through (−2, 8). What is its y-intercept?
- Write y = 4x + b and put in the point: 8 = 4 × (−2) + b.
- So 8 = −8 + b, and b = 8 − (−8) = 16.
Answer: 16
x-intercept
worked example
Where does y = −x − 12 cross the x-axis? Give the x-value as a fraction or a whole number.
- Set y = 0: −x − 12 = 0.
- So −x = 12 and x = 12 ÷ (−1) = −12.
Answer: -12
Perpendicular slope
worked example
A line has slope −3/5. What is the slope of a line perpendicular to it? Give a fraction or a whole number.
- Perpendicular slopes are negative reciprocals: flip −3/5 to −5/3 and change the sign: 5/3.
Answer: 5/3