Slope vs. Point-Slope Formula
This Grade 6 algebra skill from Yoshiwara Elementary Algebra distinguishes between slope and the point-slope formula for a line. Students learn that slope describes steepness, while the point-slope form y - y1 = m(x - x1) uses a known point and slope to write the equation of a specific line.
Key Concepts
Property They are really the same formula, but they are used for different purposes: We use the slope formula to calculate the slope of a line when we know two points on the line. That is, we know $(x 1, y 1)$ and $(x 2, y 2)$, and we are looking for $m$. The slope formula is $m = \frac{y 2 y 1}{x 2 x 1}$. We use the point slope formula to find the equation of a line. That is, we know $(x 1, y 1)$ and $m$, and we are looking for an equation like $y = mx + b$. The point slope formula is $m = \frac{y y 1}{x x 1}$.
Examples To find the slope of a line passing through $(3, 6)$ and $(5, 12)$, use the slope formula: $m = \frac{12 6}{5 3} = \frac{6}{2} = 3$. A line has a slope of $5$ and passes through $( 1, 4)$. To find its equation, use the point slope formula: $y 4 = 5(x ( 1))$, which simplifies to $y = 5x + 9$. To find the equation of a line through $(2, 1)$ and $(4, 7)$, first find the slope $m = \frac{7 1}{4 2} = 3$. Then use the point slope formula with $(2, 1)$: $y 1 = 3(x 2)$.
Explanation The slope formula finds the value of the slope, $m$, using two known points. The point slope formula uses that slope and one point to build the entire equation of the line. Think of it as finding the tool, then using the tool.
Common Questions
What is the point-slope formula for a line?
The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a known point on the line.
When do you use the point-slope formula instead of slope-intercept form?
Use point-slope form when you know the slope and a point on the line but not the y-intercept. You can then convert to slope-intercept form if needed.
How is slope different from the point-slope formula?
Slope (m) is a single number describing steepness. The point-slope formula is a full equation that specifies a unique line through a particular point.
How do you convert point-slope form to slope-intercept form?
Expand and rearrange: y - y1 = m(x - x1) becomes y = mx - mx1 + y1. Combine constants to get y = mx + b.
Where is the point-slope formula taught in Grade 6?
Slope vs. point-slope formula is covered in the Yoshiwara Elementary Algebra textbook for Grade 6.