Math · College algebra · Worked example

Find the equation of a line through two points

Find the equation of the line through (−2, 7) and (4, −2) in slope-intercept form.

Find the slope

Subtract in the same order on top and bottom: the second point minus the first.

\frac{- 2 - 7}{4 - (- 2)} = - \frac{3}{2}

Start from the point (−2, 7) with slope −3/2: y − 7 = −(3/2)(x + 2). Add 7 to both sides and expand to reach slope-intercept form.

\begin{array}{l} y = 7 - \frac{3}{2} (x + 2) \\ y = - \frac{3}{2} x + 4 \end{array}

The line crosses the y-axis at (0, 4). From any point on it, 2 units to the right the line is 3 units lower.

At x = 4 the line gives y = −2, the second point.

- \frac{3}{2} (4) + 4 = - 2

y = −(3/2)x + 4: slope −3/2 and y-intercept 4.

Find the equation of the line through (1, −3) and (5, 5).

Show the answer and explanation

The slope is (5 − (−3))/(5 − 1) = 8/4 = 2. Then y = −3 + 2(x − 1) = 2x − 5, and x = 5 gives y = 5 as it should.

\begin{array}{l} y = - 3 + 2 (x - 1) \\ y = 2 x - 5 \end{array}

Graph plots y = −(3/2)x + 4 with both points and the intercept marked. Change the slope or intercept to see which points the line leaves behind.

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