Point-Slope Formula
The Point-Slope Formula provides a method for writing the equation of a line when given its slope m and a point (x1, y1) on the line: y - y1 = m(x - x1). This formula is covered in Yoshiwara Elementary Algebra Chapter 4: Applications of Linear Equations and is essential for Grade 6 algebra students modeling real-world linear relationships. It directly connects the geometric properties of a line to its algebraic equation.
Key Concepts
Property To find an equation for the line of slope $m$ passing through the point $(x 1, y 1)$, we use the point slope formula $$\frac{y y 1}{x x 1} = m$$ or $$y y 1 = m(x x 1)$$ To Fit a Line through Two Points. 1. Compute the slope between the two points. 2. Substitute the slope and either point into the point slope formula.
Examples To find the equation for a line with slope $m=4$ passing through $(1,3)$, use the formula $y y 1 = m(x x 1)$ to get $y 3 = 4(x 1)$. To find the equation for the line passing through $(2,5)$ and $(4,11)$, first calculate the slope: $m = \frac{11 5}{4 2} = \frac{6}{2} = 3$. Then use the point slope formula with the point $(2,5)$: $y 5 = 3(x 2)$. A line passes through $( 2, 8)$ with a slope of $ 3$. To write its equation in slope intercept form, start with point slope: $y 8 = 3(x ( 2))$. This becomes $y 8 = 3(x+2)$. Distribute to get $y 8 = 3x 6$, then add 8 to both sides: $y = 3x+2$.
Explanation This formula is the most direct way to write a line's equation when you have its slope and any point it passes through. It's built directly from the definition of slope, $m = \frac{\text{rise}}{\text{run}}$.
Common Questions
What is the point-slope formula?
The point-slope formula is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a known point on the line. It is used to write the equation of a line.
When do you use point-slope form?
Use point-slope form when you know the slope of a line and one point on it, but not the y-intercept. It is especially useful for writing linear equations quickly.
How do you convert point-slope to slope-intercept form?
Expand y - y1 = m(x - x1) using the distributive property and then solve for y to get the slope-intercept form y = mx + b.
Where is the point-slope formula in Yoshiwara Elementary Algebra?
It is taught in Chapter 4: Applications of Linear Equations in Yoshiwara Elementary Algebra.
What is the difference between point-slope and slope-intercept form?
Point-slope form uses any known point (x1, y1), while slope-intercept form y = mx + b specifically uses the y-intercept (0, b). Both describe the same line.