1
Point-slope is the definition of slope, rearranged
Slope from (x₁, y₁) to a general point (x, y) is (y − y₁)/(x − x₁) = m. Multiplying up gives the form. It is not a new fact to memorise, it is “Slope” with the fraction cleared.
Lesson · Lines, Functions and Graphs · Lines
Point-Slope and Standard Form
Point-slope form writes down a line the moment you have one point and a slope, with no solving. Standard form keeps x and y symmetric, which is what systems and intercepts want. Neither replaces the other.
y − y₁ = m(x − x₁) point-slope
Ax + By = C standard
two points → find m → point-slope → simplify
either point works; the line is the same
Two more forms, and when each wins
Core ideas
What this lesson establishes
A line is pinned down by a point and a direction, or by two points. Each form below is the shortest route from one of those starting positions to an equation.
2
Either point gives the same line
Given two points you may build from whichever you prefer. The two equations look different and simplify to the same thing, which is worth seeing once so that a mismatch with a textbook answer does not read as an error.
3
Standard form is symmetric, and that is its use
Ax + By = C treats the two variables alike, so it covers vertical lines, hands you both intercepts cheaply, and is the shape Systems and Matrices needs for elimination. It hides the slope, which is the price.
The material
Two forms and the choice between them
Where each comes from, what each makes easy, and how to convert.
Definition · Point-slope form
The line through (x₁, y₁) with slope m is y − y₁ = m(x − x₁).
Theorem · Why the form is what it is
Let (x, y) be any other point on the line. By “Slope” the slope computed from (x₁, y₁) to (x, y) must equal m, so (y − y₁)/(x − x₁) = m. Multiplying both sides by (x − x₁) gives y − y₁ = m(x − x₁). The multiplication also admits the point (x₁, y₁) itself, which the fraction had to exclude.
That derivation is worth following once, because it explains the minus signs. The form contains x − x₁, so a point with a negative coordinate produces a double negative: through (−2, 1) the form is y − 1 = m(x − (−2)), that is, y − 1 = m(x + 2). Writing the substitution before simplifying is what keeps that straight.
From two points
Two points is the most common starting position, and it is handled in two moves: compute the slope by “Slope”, then feed it and either point into point-slope. Nothing else is needed.
through (-2, 1) and (4, 10)
slope m = (10 - 1)/(4 - (-2)) = 9/6 = 3/2
using (-2, 1): y - 1 = (3/2)(x + 2)
y - 1 = (3/2)x + 3
y = (3/2)x + 4
using (4, 10): y - 10 = (3/2)(x - 4)
y - 10 = (3/2)x - 6
y = (3/2)x + 4 the same line
The two routes converge, as the concept above promised. If yours does not converge with a printed answer, put both into slope-intercept form and compare there — that form is unique for a given line, so it settles the question.
Standard form
Definition · Standard form
Ax + By = C with A, B, C integers, A and B not both zero. It is conventional to clear fractions and to take A positive.
Converting into it means clearing denominators and collecting both variables on the left. From y = (3/2)x + 4: multiply by 2 to get 2y = 3x + 8, then move the x-term across for −3x + 2y = 8, and finally negate throughout so that A is positive: 3x − 2y = −8. Negating an equation changes no solutions.
Example · Checking the conversion
3x − 2y = −8 at (−2, 1): −6 − 2 = −8, true. At (4, 10): 12 − 20 = −8, true. Both original points satisfy the converted equation, which is what a conversion has to preserve.
Standard form is the only one of the three that copes with every line: x = 5 is 1·x + 0·y = 5. Point-slope needs a slope to exist and slope-intercept needs to be solvable for y, so both fail on vertical lines, while standard form does not notice the difference.
Choosing between the forms is part of the skill. Point and slope, or two points, go to point-slope. A question about shape, steepness or comparison goes to slope-intercept. Intercepts, or a system to be solved by elimination in Systems and Matrices, go to standard.
Interactive
Point-slope form
Either point gives the same line, and the lab proves it
Method
Finding the equation of a line through two points
Slope, substitute, simplify, verify with the point you did not use.
1
Compute the slope
m = (y₂ − y₁)/(x₂ − x₁), in a consistent order. If the two x-values agree the line is vertical and the answer is x = x₁; stop there.
2
Substitute into point-slope
y − y₁ = m(x − x₁) with either point. Write the substitution out, including the brackets around a negative coordinate, before touching it.
3
Simplify to the form the question wants
Distribute m, then isolate y for slope-intercept, or clear fractions and collect terms for standard form.
4
Check with the other point
Substitute the point you did not build from. It must satisfy the final equation — this is the check that catches a slope or sign error, because that point played no part in producing the answer.
Building a line
Score 0 / 0Choose the answer you can justify.
The line through (−2, 1) and (4, 10), in all three forms
Slope first, then point-slope, then the two simplified forms, then a check that uses both points.
1. slope
m = (10 - 1) / (4 - (-2)) = 9/6 = 3/2
2. point-slope, from (-2, 1)
y - 1 = (3/2)(x - (-2))
y - 1 = (3/2)(x + 2)
3. slope-intercept
y - 1 = (3/2)x + 3
y = (3/2)x + 4
4. standard
2y = 3x + 8 multiply through by 2
-3x + 2y = 8 collect on the left
3x - 2y = -8 negate so that A > 0
5. check BOTH points in the final form
(-2, 1): 3(-2) - 2(1) = -6 - 2 = -8 TRUE
( 4, 10): 3(4) - 2(10) = 12 - 20 = -8 TRUE
Step 2 is where the sign errors live. The form subtracts x₁, and x₁ is −2, so the bracket becomes x + 2. Writing x − 2 here produces a line parallel to the right one and through neither point.
Step 5 is the reason to trust the answer. The point (4, 10) was never used after step 1, so its satisfying the final equation is genuine evidence rather than a restatement of the work.
For a faded construction, use the points (−3, 5) and (1, −3). The supplied slope is −2. Build point-slope form from (−3, 5), convert it to slope-intercept and standard form, then use (1, −3) as the unused-point check. If the check fails, identify whether the slope, the double negative or the distribution caused it.
Common mistakes
1
Dropping the minus signs built into the formThe form is y − y₁ = m(x − x₁). Through (−2, 1) that is y − 1 = m(x + 2). Substituting a negative coordinate without brackets is how x + 2 becomes x − 2 and the line moves.
2
Distributing over only the first term−4(x − 2) is −4x + 8. Getting −4x − 8 shifts the line by 16 and still produces a straight, plausible answer, which is why the check with the unused point matters.
3
Treating standard form as a normal form3x − 2y = −8 and 6x − 4y = −16 are the same line. Two standard forms can differ by a constant multiple, so comparing lines is a job for slope-intercept, which is unique.
Check your understanding
✓
Finish when two points, or a point and a slope, become an equation in any requested form without hesitation.
Given six lines specified in mixed ways — two points, a point and a slope, one vertical — you should produce the requested form each time and verify with a point that was not used in the construction. That verification habit is what makes the answer yours rather than the back of the book's.
Note: Point-slope is the form that generalises furthest. Later courses write the tangent to a curve at a point in exactly this shape, because a point and a direction is all a tangent is; what changes is where the slope comes from, not the form it feeds.