Slope-Point Form

Write equations using point-slope form and a known point. Math 10-C Alberta mathematics.

Lesson 6.4 of Linear Relations and Functions in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• The Slope-Point Formula. Discover why slope-point form exists and how to use it when the y-intercept is unknown. Learn the formula, its meaning, and work through a burning candle problem from two visible points.
• Describing, Converting, and Graphing. Read information directly from a slope-point equation, convert it to slope-intercept form, and graph a line through a given point. Work through a full example from slope-point to y-intercept.
• Real-World Application — Temperature Conversion. Apply slope-point form to the Fahrenheit-to-Celsius conversion formula. Use two data points to build the equation, then substitute a real temperature to find a result.
• Parallel and Perpendicular Lines. Write equations for lines through a given point that are parallel or perpendicular to a known line. Use slope relationships and point substitution to find the y-intercept.

Inside the lesson — a free preview

Why Do We Need Another Form?

The Problem with Slope-Intercept Form. Consider a candle burning over time. The graph shows the candle's height decreasing, but the y-intercept — the point where the line crosses the y-axis — might be off the visible part of the graph.

You can still read two points from the graph and calculate the slope, but how do you write the equation without the y-intercept?

Slope-point form solves this.

The Formula

What This Formula Is Saying. The equation y - y_1 = m(x - x_1) reads:

"The change in y from the known point equals the slope times the change in x from the known point."

You can use any point on the line — you do not need the y-intercept.

Here, m is the slope and (x_1, y_1) is any known point on the line.

Worked example: The Burning Candle

A candle's height (cm) is graphed against time (min). Two visible points are (30, 20) and (60, 10).

Step 1 — Find the Slope. m = (10 - 20)/(60 - 30) = (-10)/(30) = -(1)/(3)

The candle loses 1 cm of height every 3 minutes.

Step 2 — Write in Slope-Point Form Using (30, 20). y - 20 = -(1)/(3)(x - 30)

Step 3 — Convert to Slope-Intercept Form (Optional). y - 20 = -(1)/(3)x + 10

y = -(1)/(3)x + 30

The y-intercept of 30 tells us the candle started at 30 cm tall.

Create a free Studyio account to take the full lesson with voice-over teaching, practice questions and instant feedback.