point-slope form
Write linear equations in point-slope form y-y1=m(x-x1): the fastest way to build an equation when given a point and slope or two points before converting to slope-intercept form.
Key Concepts
If you know the slope $m$ and an ordered pair $(x 1, y 1)$ of any point on the line, then you can use the point slope form to write the equation of the line: $y y 1 = m(x x 1)$.
A line has slope $ 2$ and passes through $(10, 12)$. Setup: $y 12 = 2(x 10)$. Solving gives $y = 2x + 20 + 12$, which simplifies to $y = 2x + 32$. For a line with slope 3 through $( 4, 1)$: $y 1 = 3(x ( 4))$, which simplifies to $y = 3x + 13$.
Got a point and a slope? This formula is your best friend. It’s the starting block for building the line's full equation when you don't know the y intercept right away. Just plug in what you know, solve for y, and you’re golden!
Common Questions
What is point-slope form and when do you use it?
Point-slope form is y-y1=m(x-x1), where m is the slope and (x1,y1) is any known point on the line. Use it when you are given a slope and one point, or after calculating slope from two points, before converting to another form.
How do you convert point-slope form to slope-intercept form?
Distribute the slope on the right side, then add y1 to both sides to isolate y. For y-3=2(x-1): distribute to get y-3=2x-2, then add 3 to both sides to get y=2x+1.
What is the difference between point-slope form and standard form?
Point-slope form uses a specific point and slope directly: y-y1=m(x-x1). Standard form Ax+By=C has integer coefficients with A positive and emphasizes the relationship between x and y without highlighting slope explicitly.