Slope and y-Intercept Practice Problems

What this problem type covers

Each problem gives two points with whole-number coordinates and asks for the slope and the y-intercept. Early bands have integer slopes; later ones have fractional slopes that must stay as reduced fractions.

Example problems

One example from each difficulty band, with worked solutions.

These problems are generated automatically. The answers are derived as each problem is built and are checked by tests, but mistakes are still possible — please tell us if you find one.

1. 1 · ★1
   Find the slope and the y-intercept of the line through these two points.
   (−1, 7), (0, 4)
   • ①slope = 3, y-intercept = 4
   • ②slope = −1/3, y-intercept = 4
   • ③slope = −3, y-intercept = −4
   • ④slope = −3, y-intercept = 4
   • ⑤slope = −3, y-intercept = 7
   Show solution

   ④slope = −3, y-intercept = 4

   1. Slope = (y₂ − y₁) / (x₂ − x₁) = (4 − 7) / (0 − −1) = −3
   2. Substitute one point into y = mx + b: 7 = −3·(−1) + b, so b = 4
   3. The line is y = −3x + 4
   4. So slope = −3, y-intercept = 4
2. 2 · ★2
   Find the slope and the y-intercept of the line through these two points.
   (1, 0), (2, 5)
   • ①slope = 5, y-intercept = −5
   • ②slope = 1/5, y-intercept = −5
   • ③slope = 5, y-intercept = 5
   • ④slope = −5, y-intercept = −5
   • ⑤slope = 5, y-intercept = 0
   Show solution

   ①slope = 5, y-intercept = −5

   1. Slope = (y₂ − y₁) / (x₂ − x₁) = (5 − 0) / (2 − 1) = 5
   2. Substitute one point into y = mx + b: 0 = 5·(1) + b, so b = −5
   3. The line is y = 5x − 5
   4. So slope = 5, y-intercept = −5
3. 3 · ★3
   Find the slope and the y-intercept of the line through these two points.
   (−4, −4), (−3, −2)
   • ①slope = 2, y-intercept = 5
   • ②slope = −2, y-intercept = 4
   • ③slope = 2, y-intercept = −4
   • ④slope = 2, y-intercept = 4
   • ⑤slope = 1/2, y-intercept = 4
   Show solution

   ④slope = 2, y-intercept = 4

   1. Slope = (y₂ − y₁) / (x₂ − x₁) = (−2 − −4) / (−3 − −4) = 2
   2. Substitute one point into y = mx + b: −4 = 2·(−4) + b, so b = 4
   3. The line is y = 2x + 4
   4. So slope = 2, y-intercept = 4
4. 4 · ★4
   Find the slope and the y-intercept of the line through these two points.
   (−10, 24), (−8, 21)
   • ①slope = 3, y-intercept = 9
   • ②slope = −3/2, y-intercept = 9
   • ③slope = −3/2, y-intercept = −9
   • ④slope = −3/2, y-intercept = 24
   • ⑤slope = −2/3, y-intercept = 9
   Show solution

   ②slope = −3/2, y-intercept = 9

   1. Slope = (y₂ − y₁) / (x₂ − x₁) = (21 − 24) / (−8 − −10) = −3/2
   2. Substitute one point into y = mx + b: 24 = −3/2·(−10) + b, so b = 9
   3. The line is y = (−3/2)x + 9
   4. So slope = −3/2, y-intercept = 9
5. 5 · ★5
   Find the slope and the y-intercept of the line through these two points.
   (−3, −13), (0, −6)
   • ①slope = −7, y-intercept = −6
   • ②slope = 3/7, y-intercept = −6
   • ③slope = 7/3, y-intercept = −6
   • ④slope = 7/3, y-intercept = 6
   • ⑤slope = 7/3, y-intercept = −13
   Show solution

   ③slope = 7/3, y-intercept = −6

   1. Slope = (y₂ − y₁) / (x₂ − x₁) = (−6 − −13) / (0 − −3) = 7/3
   2. Substitute one point into y = mx + b: −13 = 7/3·(−3) + b, so b = −6
   3. The line is y = (7/3)x − 6
   4. So slope = 7/3, y-intercept = −6

Where people usually go wrong

• Inverting the slope formula. It is rise over run — (y₂ − y₁) / (x₂ − x₁), not the other way round.
• Subtracting the coordinates in a different order on top and bottom. Both differences must run from the same point to the same point, or the sign flips.
• Turning a fractional slope into a rounded decimal. 2/3 is exact; 0.67 is not the answer.
• Reading the y-intercept off one of the given points. The intercept is the y value at x = 0, which is usually neither point.
• Losing the sign of the intercept when rearranging y = mx + b.