Point Slope Form

y − 2 = 3(x − 3)

What point slope form is, and what each letter means

Point slope form is one of the ways of writing the equation of a straight line, and it is the one built around a point you already know rather than around where the line crosses the y-axis.

There are only three moving parts:

• (x₁, y₁) is the point you know. Real numbers, and they go into the equation as numbers.
• m is the slope, the amount y changes for every 1 that x changes.
• x and y stay as letters. They stand for every other point on the line, which is what turns the equation into a line rather than a single dot.

That last part is the bit worth pausing on. Two of the four letters get replaced by numbers and two of them do not, and knowing which is which is most of the skill.

The point slope formula, and where the equation comes from

The point slope formula is not a separate rule you have to memorise. It is the slope formula with the fraction cleared, and seeing that once tends to be the moment it stops feeling arbitrary.

Slope between a known point (x₁, y₁) and any other point (x, y) on the same line is:

m = (y − y₁) ÷ (x − x₁)

Multiply both sides by (x − x₁) and the fraction disappears:

y − y₁ = m(x − x₁)

That is the whole derivation, and it is the one OpenStax gives in its Intermediate Algebra 2e section on finding the equation of a line. Nothing has been invented between the two lines. If you can write the slope formula you can rebuild the point slope equation from it, which is useful in an exam and more useful on a day when the formula sheet is somewhere else.

How to solve point slope form, step by step

Solving a point slope form question is the same four moves every time, whatever the numbers look like.

1. Find the slope. If you were given it, you are done. If you were given two points, subtract the y values, subtract the x values in the same order, and divide.
2. Pick a point. Any point on the line will do.
3. Substitute. The point’s numbers go in the two slots with subscripts. Leave x and y alone.
4. Stop, or tidy. The equation is already correct. Only rearrange it if the question asks for a particular form.

Step 4 is where marks get thrown away. An unsimplified point slope equation is a complete answer, and multiplying out an expression you did not need to multiply out is just another chance to drop a sign.

Point slope form examples, including negative points

Three worked examples, arranged so each one adds a wrinkle the last did not have.

A point and a slope

Slope of 3, through the point (3, 2). Put the numbers where the subscripts are and leave x and y alone:

y − 2 = 3(x − 3)

That is a finished answer. It does not need simplifying to be correct.

When the point has a negative in it

Slope of −1/3, through (1, −3). The formula subtracts y₁, and y₁ is −3, so you are subtracting a negative:

y − (−3) = −1/3(x − 1)  →  y + 3 = −1/3(x − 1)

When you are given two points instead of a slope

Find the slope first, then use either point. Take (1, 4) and (3, 10). The slope is (10 − 4) ÷ (3 − 1) = 3. Then:

• Using (1, 4): y − 4 = 3(x − 1)
• Using (3, 10): y − 10 = 3(x − 3)

Those look like different answers and they are the same line. OpenStax says plainly that you choose one point, and either will do. Multiply both out and each one lands on y = 3x + 1. So if your answer does not match the one printed in the back of the book, check whether it simplifies to the same thing before you assume you got it wrong. You can also push both through the point slope form calculator and compare what comes out.

How to graph point slope form on a grid

Point slope form is especially quick to graph when a point and a slope are already given, because the point it hands you is one you can plot straight away without working anything out first.

1. Read the point out of the equation, remembering to flip the signs.
2. Plot it.
3. From there, count the rise and the run given by the slope. A slope of 3 is 3 up and 1 across; a slope of −1/3 is 1 down and 3 across.
4. Mark the second point and draw the line through both.

No rearranging, no working out where the line crosses an axis. That is the reason the form exists.

Point slope form to standard form, and the other rewrite

Converting point slope form into standard form is four steps of tidying. Standard form is Ax + By = C, with the x and y terms together on the left. Starting from y − 2 = 3(x − 3):

1. Multiply out the bracket: y − 2 = 3x − 9
2. Move the x term across: −3x + y − 2 = −9
3. Move the constant across: −3x + y = −7
4. Multiply through by −1 so A is positive: 3x − y = 7

Tidying the other way instead gives you y = mx + b, which is the more common destination. From the same starting point it is two steps:

y − 2 = 3(x − 3)  →  y − 2 = 3x − 9  →  y = 3x − 7

Same line, three appearances, and which one you want depends on what you are about to do with it. Graphing by hand favours the point slope version, plugging into a spreadsheet favours the other.

The three ways of writing a line, side by side

Point slope form is one of three, and they are not rivals. Each one is the convenient shape for a different starting position.

Form	Written as	What you need to have	Best when
Point slope	y − y1 = m(x − x1)	One point and the slope	You measured a point somewhere other than the y-axis
Slope intercept	y = mx + b	The slope and the y-intercept	You already know where the line crosses the y-axis, or you are putting it in a spreadsheet
Standard	Ax + By = C	Any of the above, rearranged	A question asks for it, or you are solving two equations together

All three describe the same line. Converting between them is arithmetic, not a change of meaning.

Where point slope form turns up outside the classroom

Point slope form is used anywhere a measurement gives you one reading and a rate of change, which is most of the time.

• A survey or a site level. You have a height at a known chainage and a fall per metre. That is a point and a slope, and the equation gives you the height anywhere along the run.
• Anything charged at a rate. A meter reading on a known date plus a daily usage rate predicts the next reading without ever calculating a day-zero figure.
• Calculus, later on. The equation of a tangent line is written from the point of contact and the derivative at that point. That is a point and a slope, and it is why this form is worth being fluent in before you meet it again.

In each case the y-intercept is a number that was never measured and is not needed. Being able to skip it is the practical value of the form.

One condition sits under all three. A straight line only describes the situation if the rate is constant, so point slope form works when the relationship can reasonably be treated as linear. A usage rate that changes with the season, or a fall that steepens partway along a run, needs more than one line or a different model altogether.

Point slope form practice questions, with the answers

Four questions, one for each thing that tends to go wrong. Work them before you look.

1. Write the equation of the line with slope 5 through (2, 7).
2. Write the equation of the line with slope −2 through (−4, 6).
3. Write the equation of the line through (2, 3) and (6, 11).
4. An equation reads y + 5 = 4(x − 2). What point and what slope does it describe?

Show the answers

1. y − 7 = 5(x − 2)
2. y − 6 = −2(x + 4). Both signs flip here, because x₁ is negative as well as being subtracted.
3. Slope is (11 − 3) ÷ (6 − 2) = 2, so y − 3 = 2(x − 2), or y − 11 = 2(x − 6). Both are right.
4. Slope 4, through the point (2, −5). Not (2, 5).

Frequently asked questions

What is point slope form used for?

Writing the equation of a line when you know a point on it and how steep it is, which is the situation you are usually in. Measurements give you a point and a rate. They rarely give you the y-axis crossing, so a form that needs the crossing makes you calculate something you were never asked for.

When do you use point slope form instead of another form?

Use it when what you have is a point and a slope, and especially when the point is not on the y-axis. If you happen to know where the line crosses the y-axis, the other form is fewer steps. If a question asks for the answer in a particular form, write it in point slope form first anyway and rearrange at the end; the rearranging is easier than trying to find the intercept first.

Can a vertical line be written in point slope form?

No. A vertical line has no slope to put in, because the run between any two of its points is zero and you cannot divide by zero. There is nothing to substitute for m. A vertical line is written as x = a number instead, so a vertical line through (1.5, 4) is x = 1.5. This is not a gap in the form, it is a property of vertical lines, and the same limit applies to every equation of a line that contains an m.

How do you find the slope if you are only given two points?

Subtract the y values, subtract the x values in the same order, and divide the first result by the second. Keeping the order the same in both subtractions is the part people get wrong; swapping one but not the other flips the sign of the answer. Once you have m, drop it and either of the two points into the formula.

Is an unsimplified point slope equation a complete answer?

Yes, unless the question asks for a specific form. y − 2 = 3(x − 3) describes exactly the same line as 3x − y = 7, and neither is more finished than the other. Read the instruction rather than assuming the tidiest looking version is the one wanted.