Point-Slope and Slope-Intercept Forms of a Line

The point-slope form builds a line's equation from one point on the line and its slope: y − y1 = m(x − x1). The slope-intercept form builds it from the slope and the y-intercept: y = mx + c, where c is where the line crosses the y-axis.

These two forms matter because they match the two most common situations in problems: either you know a point and the steepness, or you know the steepness and where the line starts on the y-axis. Almost every line question in school exams is solved by reaching for one of these two forms first.

Key Concepts

• Point-slope ingredients: a known point (x1, y1) on the line and the slope m. Nothing else is needed.
• Slope-intercept ingredients: the slope m and the y-intercept c, the value of y when x = 0.
• Equivalence: point-slope can always be rearranged into slope-intercept by expanding and isolating y.
• Reading off values: in y = mx + c the numbers m and c are visible directly — no solving required.
• Units: m has units of y-per-x; c has the same units as y.

Formulas

• Point-slope: y − y1 = m(x − x1), for a line through (x1, y1) with slope m.
• Slope-intercept: y = mx + c, where m is the slope and c is the y-intercept.
• Conversion: expanding point-slope gives y = mx + (y1 − mx1), so c = y1 − mx1.

Assumptions: the line is not vertical (a vertical line has no finite slope m, so neither form applies — its equation is simply x = k). Limits: point-slope needs a genuinely known point on the line; guessing a point off the line gives a wrong equation.

Worked Example

Write the equation of the line through P(3, −2) with slope 4, in both forms.

1. Point-slope: substitute x1 = 3, y1 = −2, m = 4 into y − y1 = m(x − x1), giving y − (−2) = 4(x − 3), i.e. y + 2 = 4(x − 3).
2. Expand the right side: y + 2 = 4x − 12.
3. Isolate y: y = 4x − 12 − 2 = 4x − 14.
4. So the slope-intercept form is y = 4x − 14, with slope 4 and y-intercept −14.
5. Verify: at x = 3, y = 12 − 14 = −2, which is point P. Correct.

Common Misconceptions

• "In y = mx + c, c is the x-intercept." No — c is the y-intercept (where x = 0). The x-intercept needs a separate calculation.
• "Point-slope only works if the point is the intercept." Any point on the line works — that is the whole advantage of the form.
• "y + 2 = 4(x − 3) is not a final answer." Point-slope form is a perfectly valid final equation; converting to y = mx + c is optional unless asked.

Real-World Applications

• Phone plans (fixed charge + per-minute rate) are written directly as y = mx + c.
• Scientists fit trend lines to data and read growth rate (m) and baseline (c) straight off.
• Taxi fares with a base fare plus per-km charge follow slope-intercept form.
• Engineers describe calibration curves of sensors as y = mx + c.

Point-slope form needs a point and a slope; slope-intercept form needs a slope and a y-intercept — and the two convert into each other in one line of algebra. Pick whichever matches the data you are given, and the line's equation writes itself.