Point-Slope Form
Slope-intercept form needs the one point almost no question gives you. Point-slope form takes whichever point you were actually handed, which is why it is the form to reach for first.
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Almost every question about lines gives you a point and a slope, or two points. Almost none of them gives you the y-intercept. Yet y = mx + b is usually the only form people can write down, so they spend a line and a half solving for a b nobody asked about.
Point-slope form skips that. Drag the dot below and watch the equation follow it.
Drag the dot anywhere. The equation rewrites itself around whatever point you land on - the line never has to pass through the y-axis at a whole number.
Where the formula comes from
Slope is rise over run between any two points on the line. Fix one of them at a point you know, (x₁, y₁), and let the other be a general point (x, y). Then
m = (y - y₁)/(x - x₁)
Multiply both sides by x - x₁ and the fraction clears:
y - y₁ = m(x - x₁)
That is the whole derivation. Point-slope form is not a new idea - it is the slope formula with the division undone, which is why every line satisfies it and why it works from any point on the line.
Both subtractions are distances from your point
The two minus signs are doing the same job on different axes: how far across from x₁, and how far up from y₁. Read that way, the signs stop being something to memorise.
They do bite when a coordinate is negative. A line of slope 2 through (-3, 4) is
y - 4 = 2(x + 3)
because subtracting -3 is adding 3. Getting a plus sign there is not a mistake; writing x - 3 is.
Move the point in the figure into the third quadrant and watch both signs flip. It is the one part of this form that generates wrong answers, and thirty seconds of watching it fixes it better than a rule.
Two points, one line
Given (1, 1) and (4, 7): find the slope first, then drop either point in.
m = (7 - 1)/(4 - 1) = 6/3 = 2
Using the first point gives y - 1 = 2(x - 1). Using the second gives y - 7 = 2(x - 4). They look nothing alike and they are the same line - multiply either out and you get y = 2x - 1.
That is worth trusting rather than doubting. If your answer disagrees with the back of the book, rearrange both before assuming you are wrong.
Turning it into slope-intercept form
When a question does want y = mx + b, the conversion is two moves: expand the bracket, then get y alone.
Starting from y - 3 = 2(x - 1): expanding gives y - 3 = 2x - 2, and adding 3 to both sides gives y = 2x + 1.
The b you never had to think about turns up at the end, which is the point. Point-slope did the work; slope-intercept is just the format.
Try one
The commonest slip here is expanding 2(x - 1) to 2x - 1. The 2 has to reach both terms inside the bracket, which is the distributive property doing its usual job.
Which form for which question
| you are given | reach for |
|---|---|
| a point and a slope | point-slope |
| two points | slope from them, then point-slope |
| the y-intercept and a slope | slope-intercept |
| an equation to graph quickly | slope-intercept |
| a line parallel or perpendicular to another, through a point | point-slope |
Point-slope is the form for writing a line; slope-intercept is the form for reading one. Most problems need both, in that order.
Common questions
- What is point-slope form?
- The equation y - y1 = m(x - x1), where m is the slope and (x1, y1) is any single point the line passes through. It is a direct translation of the slope formula: the rise from the known point over the run from the known point is always m, and multiplying both sides by that run clears the fraction.
- When should I use point-slope form instead of slope-intercept?
- Whenever the point you are given is not the y-intercept - which is most of the time. Slope-intercept form needs b, so if you start from a random point you have to solve for b first. Point-slope lets you write the answer down immediately and rearrange afterwards, if the question even asks you to.
- How do I find the equation of a line through two points?
- Work out the slope first: the difference in y over the difference in x. Then pick either point - genuinely either, they give the same line - and drop it into point-slope form. Rearranging is only needed if the question asks for a particular form.
- Why are the signs subtractions in point-slope form?
- Because both are differences measured from the known point: how far across from x1, and how far up from y1. When a coordinate is negative the subtraction turns into an addition, which is where most errors happen - a point at (-3, 4) gives y - 4 = m(x + 3), and that plus sign is correct.
- Does it matter which of the two points I use?
- No. The two equations look different and describe exactly the same line - rearrange both into slope-intercept form and they come out identical. Most teachers accept either, but if a question specifies a form, rearranging settles any argument.
Now do it yourself, properly
Reading someone else's working is not the same as being able to do it. The Coddy math course puts you on a board that checks every move you make, so you find out where you actually stand.
Want to run this on your own numbers?
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