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How to find the equation of a line from two points

The rule: slope first, intercept second

For points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , first calculate the slope $m = (y_{2} - y_{1}) / (x_{2} - x_{1})$ . Then put one point into $y = m x + b$ and solve for $b$ . Finally, substitute both original points into the finished equation. If both work, the signs and fraction are correct.

Problem

Write each line in fully simplified slope-intercept form $y = m x + b$ . Example 1. The line passes through $(3, 1)$ and $(6, 4)$ . Example 2. The line passes through $(- 6, 3)$ and $(6, 5)$ . Example 3. The line passes through $(- 6, 0)$ and $(0, - 5)$ .

Answer

For

(3, 1)

and

(6, 4)

, the line is

y = x - 2

. For

(- 6, 3)

and

(6, 5)

, it is

y = \frac{1}{6} x + 4

. For

(- 6, 0)

and

(0, - 5)

, it is

y = - \frac{5}{6} x - 5

Each equation was checked against both points used to construct it.

Step-by-step solution

Example 1: an integer slope

Use the slope formula:

m = \frac{4 - 1}{6 - 3} = \frac{3}{3} = 1

Now substitute $(3, 1)$ into $y = m x + b$ :

1 = 1 (3) + b

so $b = - 2$ . Therefore

y = x - 2

A quick check with the second point gives $4 = 6 - 2$ .

Example 2: a positive fractional slope

The horizontal change is $12$ and the vertical change is $2$ :

m = \frac{5 - 3}{6 - ( - 6 )} = \frac{2}{12} = \frac{1}{6}

Substitute $(6, 5)$ :

5 = \frac{1}{6} (6) + b = 1 + b

Thus $b = 4$ and

y = \frac{1}{6} x + 4

Checking $x = - 6$ gives $y = - 1 + 4 = 3$ .

Example 3: a negative fractional slope

Use $(- 6, 0)$ and $(0, - 5)$ :

m = \frac{- 5 - 0}{0 - ( - 6 )} = - \frac{5}{6}

The point $(0, - 5)$ gives the intercept immediately: $b = - 5$ . Hence

y = - \frac{5}{6} x - 5

At $x = - 6$ , this gives $y = 5 - 5 = 0$ , as required.

How to check yourself

A line with a positive slope rises from left to right; a negative slope falls. Then substitute both given points. This catches the two most common errors: reversing only one subtraction in the slope formula and losing the sign of $b$ .

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