Point Slope Form Calculator
Result
Point-slope form
- Slope-intercept form
- y = 2x + 1
- y-intercept
- 1.000000
A point slope form calculator writes the equation of a line when one point on it and its slope are known. The answer comes back in two arrangements: the point-slope form, which keeps the point visible as the numbers it was built from, and the slope-intercept form, which multiplies it out so the y intercept can be read off directly. The default is a slope of 2 through the point (1, 3), giving y − 3 = 2(x − 1), which is y = 2x + 1 once rearranged. Everything the line does is already visible in the first arrangement, which is the reason for preferring it: the slope is the number in front, and the point's coordinates sit where they were entered. A fractional slope is a legitimate input and is written as a fraction rather than rounded, because the equation is an exact object and a decimal would only be one spelling of one of its numbers.
Ten slopes and points, and the equations they produce
| Slope | x₁ | y₁ | Point-slope form |
|---|---|---|---|
| 2 | 1 | 3 | y - 3 = 2(x - 1) |
| 1 | 0 | 0 | y = x |
| 1 | 5 | 0 | y = (x - 5) |
| -1 | 2 | 5 | y - 5 = -(x - 2) |
| -1 | 0 | 4 | y - 4 = -x |
| 0 | 1 | 3 | y - 3 = 0 |
| 0.5 | 4 | 1 | y - 1 = (1/2)(x - 4) |
| 3 | 0 | -2 | y + 2 = 3x |
| -2 | -1 | -3 | y + 3 = -2(x + 1) |
| 4 | 2 | 2 | y - 2 = 4(x - 2) |
Each row takes a slope and a point and shows the equation that comes out. The rows were chosen to cover the writing conventions rather than to be arbitrary, since those are what a reader is checking against: the second row has both coordinates zero, which drops the bracket entirely, and the third has a zero height, which leaves the left-hand side as a bare y. The sixth row has a slope of zero and prints a plain 0 on the right. The seventh carries a fractional slope, which is bracketed, and the ninth has negative coordinates, which flip both signs at once. Between them these are the cases a written answer gets wrong, and the signed coordinates are what make the second and ninth rows worth setting side by side. Every cell is a number or notation, so the table is identical in all ten languages the site serves.
Formula
y − y₁ = m(x − x₁) slope-intercept form: y = mx + (y₁ − m·x₁)
- m
- The slope: how steeply the line rises or falls, as a signed number. It is the coefficient in front of the bracket on the right-hand side, and the coefficient of x once the equation is rearranged. A slope of 0 makes the line horizontal, which is the case where the calculator writes a plain 0 instead of a bracket multiplied by nothing.
- x₁, y₁
- The one known point. Only one is needed here: the slope fixes the direction and the point fixes the position, and together those determine the line. The subscripts are the ones the formula uses rather than an index into a list — there is no second point on this page.
- y intercept
- Where the line crosses the y axis, which is the constant left over after the equation is rearranged. It comes out of the same three inputs: the line passes through the point given, so its height at x equal to zero is that point's height minus the rise the slope would have produced getting there.
- x, y
- The two variables the equation relates. Neither is given a value: the answer is a rule applying to every point on the line at once, which is why the main reading is text rather than a single number.
Use this to write down a line when its steepness and one point on it are known: from a rate and a starting value, from a tangent's slope at a known location, or from two points after the slope between them has been worked out. Going the other way from two points to a slope is the slope page; using a finished equation to predict a value further along the line is the interpolation page.
Worked examples
A slope of 2 through (1, 3)
- Put the slope and the point into the template: y − y₁ = m(x − x₁)
- The left-hand side becomes y − 3 and the bracket becomes (x − 1)
- The point-slope form is y − 3 = 2(x − 1)
- Multiply the bracket out: 2(x − 1) = 2x − 2
- Move the −3 across: y = 2x − 2 + 3 = 2x + 1
- The slope-intercept form is y = 2x + 1, so the y intercept is 1
The default case, and the one to read against the inputs: the two numbers inside the brackets are the coordinates that were typed in, with their signs flipped by the subtraction. That is what the point-slope arrangement is for — it keeps the given point readable instead of folding it into a constant.
A slope of −1, where the bracket must stay
- The template gives y − 5 = m(x − 2)
- A slope of −1 is written as a bare minus sign rather than as −1, the same way a coefficient of 1 is left off
- The bracket stays: writing −x − 2 instead would be a different line
- The point-slope form is y − 5 = −(x − 2)
- Multiply out: −(x − 2) = −x + 2, then y = −x + 2 + 5
- The slope-intercept form is y = −x + 7 and the y intercept is 7
The row most likely to go wrong. A slope of 1 and a slope of −1 are both written without a visible number, but the minus sign has to stay outside the bracket: dropping the bracket turns −(x − 2) into −x − 2, a sign error in the middle of an equation that otherwise looks perfectly normal.
A fractional slope
- A slope of 0.5 is kept as the exact fraction 1/2 rather than written as a decimal
- The template gives y − 1 = (1/2)(x − 4)
- The fraction is bracketed, because 1/2(x − 4) would read as one divided by twice the bracket
- The same bracketing is repeated in the slope-intercept form: y = (1/2)x − 1
- The y intercept is −1
The case that decides two conventions at once. Fractions are kept exact so the equation stays an equation rather than an approximation, and a fractional coefficient is bracketed so that it stays attached to the bracket and not to the variable. Both conventions match the ones the other algebra pages use for the same kind of number.
Limitations
The slope and both coordinates are limited to a magnitude of one million. Fractional inputs are printed as fractions when the denominator divides evenly and is at most one thousand, and as six decimals otherwise — so 0.5 comes out as 1/2 while a decimal with a long repeating expansion falls back on rounding. Slope and coordinates are accepted as decimals only; the page does not take a fraction typed as text, so 1/3 has to be entered as the decimal it rounds to. A vertical line cannot be written in this form at all: its slope is undefined, and no value of m produces it, so a vertical line through the given point is simply not one of the lines this page can describe. What the page does not do is find the slope for you from two points, or tell you where the line crosses the x axis.
Frequently asked questions
- What is point slope form?
- A way of writing a linear equation that keeps one known point on the line visible: y minus the point's height equals the slope times x minus the point's position. Any point and any slope can be written this way, and reading the equation back tells you which point and which slope produced it.
- Why is it useful if the slope-intercept form is simpler?
- Because it is built directly from what is known. The slope-intercept form is tidier to plot from, but getting it requires multiplying out and collecting terms, and any arithmetic slip hides inside a constant. The point-slope form holds the given numbers where they were entered, so it can be checked at a glance.
- What does the y intercept tell me?
- Where the line meets the y axis, which is its height when x is zero. It follows from the same three inputs: start at the known point and undo the rise the slope would have produced getting there from the axis. If the known point is already on the axis, the intercept is simply its height.
- Why is a slope of 1 written without a number?
- For the same reason a coefficient of 1 is left off anywhere else: writing 1(x − 2) adds a symbol that carries no information. A slope of −1 keeps its minus sign but still drops the digit, and that is where the bracket has to stay — the minus belongs outside it, and removing the bracket changes the line.
- Why are fractional slopes bracketed?
- To keep the fraction attached to the bracket rather than to the x. Written without brackets, 1/2(x − 4) reads as one divided by twice the bracket, which is a different expression. Bracketing it gives (1/2)(x − 4), and the multiplication order becomes unambiguous.
- Can a vertical line be written this way?
- No. A vertical line has no slope — its rise is real while its run is zero — so there is no value of m that produces it, and the point-slope form cannot describe it. Such a line is written as x equal to a constant instead, and that form is outside what this page handles.
References
- Point-Slope Form — the arrangement this page starts from, and the reason a slope and a single point are enough — Wolfram MathWorld (United States)
- Linear Equation — the general form the two outputs are both special cases of, and the standard form the answer can be rearranged into — Wolfram MathWorld (United States)
- Slope — the number that fixes the line's direction, and the quantity that has no value at all for a vertical line — Wolfram MathWorld (United States)