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Mathematics, three units · Ages 16–17

Finding the Equation of a Line Through Two Points

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For 11th-grade students. Calculate the slope m and the y-intercept n for each pair of points, write the final line equation in the form y = mx + n, and verify your answer with the second point.

  1. Find the equation of the line passing through (0, 3) and (2, 7).

  2. Find the equation of the line passing through (1, 4) and (3, 10).

  3. Find the equation of the line passing through (2, 8) and (5, 2).

  4. Find the equation of the line passing through (-1, -5) and (3, 3).

  5. Find the equation of the line passing through (-4, 7) and (2, 7).

  6. A gym membership charges a fixed registration fee plus a set monthly fee. After 2 months, a client paid $70. After 5 months, the client paid $145. Write a linear equation for total cost y after x months.

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Finding the Equation of a Line Through Two Points

Answer key

  1. Find the equation of the line passing through (0, 3) and (2, 7).

    m = (7 - 3) / (2 - 0) = 2. Since x = 0 at (0, 3), n = 3. Equation: y = 2x + 3.

  2. Find the equation of the line passing through (1, 4) and (3, 10).

    m = (10 - 4) / (3 - 1) = 6 / 2 = 3. Substitute (1, 4): 4 = 3(1) + n, so n = 1. Equation: y = 3x + 1.

  3. Find the equation of the line passing through (2, 8) and (5, 2).

    m = (2 - 8) / (5 - 2) = -6 / 3 = -2. Substitute (2, 8): 8 = -2(2) + n, so n = 12. Equation: y = -2x + 12.

  4. Find the equation of the line passing through (-1, -5) and (3, 3).

    m = (3 - (-5)) / (3 - (-1)) = 8 / 4 = 2. Substitute (3, 3): 3 = 2(3) + n, so n = -3. Equation: y = 2x - 3.

  5. Find the equation of the line passing through (-4, 7) and (2, 7).

    m = (7 - 7) / (2 - (-4)) = 0 / 6 = 0. The line is horizontal with n = 7. Equation: y = 7.

  6. A gym membership charges a fixed registration fee plus a set monthly fee. After 2 months, a client paid $70. After 5 months, the client paid $145. Write a linear equation for total cost y after x months.

    Points are (2, 70) and (5, 145). Slope m = (145 - 70) / (5 - 2) = 75 / 3 = 25. Substitute (2, 70): 70 = 25(2) + n, so n = 20. Equation: y = 25x + 20.