Equation of a Line using Two Points Worksheets - Free Printable

To find the equation of a line passing through two given points, we use the Two-Point Form of a linear equation:

> Two-Point Form:
> \[
> y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)
> \]
> where \((x_1, y_1)\) and \((x_2, y_2)\) are the two given points.

Alternatively, after finding the slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\), you can use point-slope form:
\[
y - y_1 = m(x - x_1)
\]

Then, convert to slope-intercept form (\(y = mx + b\)) or standard form (\(Ax + By = C\)) as needed. For this worksheet, we’ll generally present the answer in slope-intercept form unless otherwise specified.

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Let’s solve each problem one by one.

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1) Points: (–6, –2) and (–9, –3)

- Slope \(m = \frac{-3 - (-2)}{-9 - (-6)} = \frac{-1}{-3} = \frac{1}{3}\)

- Use point-slope form with point (–6, –2):

\[
y - (-2) = \frac{1}{3}(x - (-6))
\Rightarrow y + 2 = \frac{1}{3}(x + 6)
\]

- Simplify:

\[
y = \frac{1}{3}x + 2 - 2 = \frac{1}{3}x
\]

✔ Answer: \(y = \frac{1}{3}x\)

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2) Points: (1, 14) and (4, 11)

- Slope \(m = \frac{11 - 14}{4 - 1} = \frac{-3}{3} = -1\)

- Use point-slope with (1, 14):

\[
y - 14 = -1(x - 1)
\Rightarrow y = -x + 1 + 14 = -x + 15
\]

✔ Answer: \(y = -x + 15\)

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3) Points: (9, 10) and (5, 4)

- Slope \(m = \frac{4 - 10}{5 - 9} = \frac{-6}{-4} = \frac{3}{2}\)

- Use point-slope with (9, 10):

\[
y - 10 = \frac{3}{2}(x - 9)
\]

- Simplify:

\[
y = \frac{3}{2}x - \frac{27}{2} + 10 = \frac{3}{2}x - \frac{27}{2} + \frac{20}{2} = \frac{3}{2}x - \frac{7}{2}
\]

✔ Answer: \(y = \frac{3}{2}x - \frac{7}{2}\)

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4) Points: (7, 0) and (8, 9)

- Slope \(m = \frac{9 - 0}{8 - 7} = \frac{9}{1} = 9\)

- Use point-slope with (7, 0):

\[
y - 0 = 9(x - 7) \Rightarrow y = 9x - 63
\]

✔ Answer: \(y = 9x - 63\)

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5) Points: (8, –1) and (10, 1)

- Slope \(m = \frac{1 - (-1)}{10 - 8} = \frac{2}{2} = 1\)

- Use point-slope with (8, –1):

\[
y + 1 = 1(x - 8) \Rightarrow y = x - 8 - 1 = x - 9
\]

✔ Answer: \(y = x - 9\)

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6) Points: (–4, –8) and (3, 6)

- Slope \(m = \frac{6 - (-8)}{3 - (-4)} = \frac{14}{7} = 2\)

- Use point-slope with (–4, –8):

\[
y + 8 = 2(x + 4) \Rightarrow y = 2x + 8 - 8 = 2x
\]

✔ Answer: \(y = 2x\)

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7) Points: (10, –3) and (2, 5)

- Slope \(m = \frac{5 - (-3)}{2 - 10} = \frac{8}{-8} = -1\)

- Use point-slope with (10, –3):

\[
y + 3 = -1(x - 10) \Rightarrow y = -x + 10 - 3 = -x + 7
\]

✔ Answer: \(y = -x + 7\)

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8) Points: (–5, 6) and (–6, 8)

- Slope \(m = \frac{8 - 6}{-6 - (-5)} = \frac{2}{-1} = -2\)

- Use point-slope with (–5, 6):

\[
y - 6 = -2(x + 5) \Rightarrow y = -2x - 10 + 6 = -2x - 4
\]

✔ Answer: \(y = -2x - 4\)

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9) Points: (–2, 5) and (7, 2)

- Slope \(m = \frac{2 - 5}{7 - (-2)} = \frac{-3}{9} = -\frac{1}{3}\)

- Use point-slope with (–2, 5):

\[
y - 5 = -\frac{1}{3}(x + 2)
\]

- Simplify:

\[
y = -\frac{1}{3}x - \frac{2}{3} + 5 = -\frac{1}{3}x - \frac{2}{3} + \frac{15}{3} = -\frac{1}{3}x + \frac{13}{3}
\]

✔ Answer: \(y = -\frac{1}{3}x + \frac{13}{3}\)

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## ✔ Final Answers:

1) \(y = \frac{1}{3}x\)
2) \(y = -x + 15\)
3) \(y = \frac{3}{2}x - \frac{7}{2}\)
4) \(y = 9x - 63\)
5) \(y = x - 9\)
6) \(y = 2x\)
7) \(y = -x + 7\)
8) \(y = -2x - 4\)
9) \(y = -\frac{1}{3}x + \frac{13}{3}\)

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💡 Tip: Always check your work by plugging one of the original points into your final equation to verify it satisfies the equation.

Let me know if you’d like these converted to standard form or graphed!