Point-Slope Form

What You’ll Learn

In this lesson you’ll learn the point-slope form of a linear equation, how to write it from a point and slope, and how to convert it to other forms.

The Concept

Point-slope form is a way to write the equation of a line when you know one point on the line and the slope. The formula is:

$y - y_1 = m(x - x_1)$

Where:

• m = slope
• (x₁, y₁) = any point on the line

This form is useful when you have a point and slope (common in word problems about rates or changes).

To graph: start at the given point (x₁, y₁), use slope m to find another point, draw the line.

To convert to slope-intercept (y = mx + b): distribute m, then add y₁ to both sides.

Example: Point (3, 5), slope m = 2

y − 5 = 2(x − 3) y − 5 = 2x − 6 y = 2x − 1

Point-slope → slope-intercept: y = 2x − 1

Worked Example

Write the equation in point-slope form for a line with slope −3 passing through point (4, 1).

1. Plug in m = −3, (x₁, y₁) = (4, 1):

   y − 1 = −3(x − 4)

2. Convert to slope-intercept (optional):

   y − 1 = −3x + 12 y = −3x + 13

Another: Slope 1/2, point (−2, 7)

y − 7 = (1/2)(x − (−2)) y − 7 = (1/2)(x + 2)

Real-World Application

Point-slope form is great for situations where you know a starting point and a rate of change:

• Savings: You start with 200 dollars and add 50 dollars per week. Slope = 50, point (0 weeks, 200 dollars) → y − 200 = 50(x − 0) → y = 50x + 200
• Temperature drop: Starts at 72°F and drops 4°F per hour. Slope = −4, point (0 hours, 72) → y − 72 = −4(x − 0)
• Cost increase: Item costs 25 dollars now and rises 2 dollars per year. Slope = 2, point (0 years, 25) → y − 25 = 2(x − 0)

This form helps model growth, decay, or changes starting from a known value.

You’ve Got This

Point-slope form is perfect when you know “where you start” (the point) and “how fast it’s changing” (the slope). Plug them in, distribute if you need y = mx + b, and check by plugging in the original point. Practice with rates of change (pay raises, costs, temperature) and you’ll find it useful for real predictions.

Quiz

What is the point-slope form for slope 3 and point (2, 5)?

• A.y − 5 = 3(x − 2)
• B.y = 3x + 5
• C.y − 2 = 3(x − 5)
• D.y − 5 = 3x − 2

A line has slope −4 and passes through (1, 8). Which equation?

• A.y − 8 = −4(x − 1)
• B.y = −4x + 8
• C.y − 1 = −4(x − 8)
• D.y − 8 = −4x − 1

You start with 100 dollars in savings and add 25 dollars per week. Which point-slope equation?

• A.y − 100 = 25(x − 0)
• B.y = 25x + 100
• C.y − 0 = 25(x − 100)
• D.y − 100 = 25x

Convert y − 3 = 2(x + 4) to slope-intercept form.

• A.y = 2x + 11
• B.y = 2x − 5
• C.y = 2x + 3
• D.y = 2x − 11

Write the equation of a line through $(3, -1)$ and $(5, 3)$.

• A.$y + 1 = 2(x - 3)$
• B.$y - 3 = 2(x - 5)$
• C.$y + 1 = \frac{1}{2}(x - 3)$
• D.$y - 1 = 2(x + 3)$