HighSchoolQ

➕ Algebra 1 · Linear Functions

Point-Slope Form

Write a line’s equation as y − y_1 = m(x − x_1) from a point and a slope or from two points, read the point correctly, and convert to y = mx + b.

Grades 9, 10⏱ 14 min✔ AI-checkedCCSS F-LE.A.2CCSS A-CED.A.2CCSS F-IF.C.7a
About standards and checking

Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Three blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-07.

  • F-LE.A.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • A-CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • F-IF.C.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima.

By the end you can…

  • Read the slope and a point on the line from an equation written as y − y1 = m(x − x1), including the sign trap when x1 or y1 is negative.
  • Write the equation of a line in point-slope form from a point and a slope, or from two points.
  • Convert point-slope form to slope-intercept form, and graph a line directly from point-slope form.
  • Use point-slope form to model a real situation where you know one reading and a rate of change.

Lesson (school style)

Point-slope form is a quick way to write the equation of a line when you know the slope and one point on the line, even if that point is not on the y-axis. It looks like y − y1 = m(x − x1). You put in the slope and the coordinates of the point, and the equation is done. There is no need to solve for b first. This lesson covers reading the form (and its sign trap), writing it from a point and a slope or from two points, converting it to slope-intercept form, and using it in a real situation.

The form y − y_1 = m(x − x_1)

A line with slope m that passes through the point (x1, y1) has the equation

y − y1 = m(x − x1)

Here is what each part means.

  • m is the slope, a number.
  • (x1, y1) is one known point on the line. The small 1 is a subscript. It only labels these as the coordinates of the known point, and it does not multiply anything. So x1 and y1 are numbers. Read x1 as “x sub 1.”
  • x and y are the variables. They stay as letters, because they stand for the coordinates of any point on the line.

For example, the line with slope 2 through the point (3, 5) is y − 5 = 2(x − 3). Some books write the same equation as y = m(x − x1) + y1, with y1 moved to the right side. In that version the x part still flips, but y1 is added, so read it with its own sign: y = 2(x + 3) − 5 has x1 = −3 and y1 = −5, so the line passes through (−3, −5). The full “flip the sign” rule, for both x and y, is for the form y − y1 = m(x − x1), where both coordinates are subtracted.

The form needs a slope that is a number, so it cannot describe a vertical line (see the two-points section).

Where the form comes from

Take a line with slope m through the known point (x1, y1), and let (x, y) be any other point on the line. Any two points on a line give the same slope, so

m = (y − y1) ÷ (x − x1), for x ≠ x1.

Multiply both sides by (x − x1), which is allowed because it is not 0 here. You get y − y1 = m(x − x1). So point-slope form is the slope formula with the division cleared. The known point itself fits too: putting in x = x1 and y = y1 gives 0 = m × 0, which is true.

You can also read the equation as a sentence. The left side, y − y1, is the rise from the known point. The factor x − x1 is the run from it. So the equation says rise = slope × run.

Reading the slope and a point (flip the sign)

In the form y − y1 = m(x − x1), the slope is the number multiplying the parenthesis, with the sign in front of it. The point comes from the two subtractions. The form has a minus sign built in, so the coordinate is the opposite of the sign you see next to x or y.

  • A minus sign gives a positive coordinate: x − 3 means x1 = 3.
  • A plus sign gives a negative coordinate, because adding is the same as subtracting a negative: x + 3 is x − (−3), so x1 = −3. In the same way, y + 4 is y − (−4), so y1 = −4.

For example, y + 8 = −(1/2)(x + 6) has slope −1/2 and passes through (−6, −8). The slope keeps its own sign. The signs inside are a separate matter, and they only tell you about the point.

If a minus sign stands alone in front of the parenthesis, as in y − 2 = −(x + 5), the slope is −1. With no number and no sign, the slope is 1.

If x or y appears alone, that coordinate is 0. For example, y − 7 = 3x means y − 7 = 3(x − 0), so the point is (0, 7).

Writing an equation from a point and a slope

  1. Write down the slope m and the point (x1, y1).
  2. Put them into y − y1 = m(x − x1). Put parentheses around any negative number as you substitute, so a minus sign is not lost.
  3. Simplify the double negatives: y − (−2) becomes y + 2, and x − (−6) becomes x + 6.

You do not have to solve for b. The equation is finished once the numbers are in. Keep a fractional slope as a fraction, and leave the parentheses alone unless the question asks for slope-intercept form.

Writing an equation from two points

The form needs a slope, so with two points find the slope first.

  1. Use the slope formula, m = (y2 − y1) ÷ (x2 − x1), and subtract in the same order on top and on the bottom.
  2. Choose either point to be the known point (x1, y1). The 1 here just means “the known point,” so it does not have to be the first point you used in the slope formula.
  3. Write y − y1 = m(x − x1).

Both choices give a correct equation. The two equations look different, but they describe the same line. Checking your equation with the other point is a good habit.

There are two special cases. If the run is 0 (both points have the same x), the line is vertical and its slope is undefined, so it has no point-slope equation. Write x = a instead. If the rise is 0, then m = 0, and y − y1 = 0(x − x1) simplifies to y = y1, a horizontal line.

Converting to slope-intercept form

To change point-slope form into y = mx + b:

  1. Distribute the slope to both terms inside the parentheses, including the sign of the second term.
  2. Move the number next to y to the right side with the opposite operation. If the left side is y − 3, add 3 to both sides. If it is y + 3, subtract 3 from both sides.

In symbols, y = m(x − x1) + y1 = mx + (y1 − m × x1), so b = y1 − m × x1.

A non-vertical line has endless point-slope equations, because every point on it can be the known point. A non-vertical line has only one equation of the form y = mx + b, because its m and b are fixed numbers. So converting both equations to slope-intercept form is a good way to check whether they describe the same line.

Graphing from point-slope form

You can graph straight from the equation. Read the slope and the point. Plot the known point (x1, y1), which is not necessarily (0, b). From it, use rise over run to find more points, and draw the line through them, extending it in both directions. To see where the line crosses the y-axis, put x = 0 into the equation and solve for y. That y-value is b, and the point is (0, b).

Point-slope form in real life

In a story, m is the rate of change (the “per” number), and the point (x1, y1) is one known reading, such as an amount at a certain time. This is where point-slope form shines: you often know one reading and a steady rate, but not the starting amount.

Write the equation, then put in any x to predict y. If you want the starting value, put in x = 0 (or convert to slope-intercept form and read b), as long as x = 0 makes sense in the story.

Give units (“10 liters per minute”), and remember that a story may fit only part of the line. A tank cannot hold a negative amount of water, for example.

Key words

Point-slope form
A way to write the equation of a line as y − y1 = m(x − x1), where m is the slope and (x1, y1) is a known point on the line.
Known point
The point (x1, y1) that you put into the form. Any point on the line will do.
Subscript
A small number written below and to the right of a letter, like the 1 in x1. It labels a particular value. It is not an exponent, and it does not multiply anything.
Slope
How steep a line is and which way it tilts: rise divided by run. In point-slope form it is m.
Slope-intercept form
A way to write a line’s equation as y = mx + b, where m is the slope and b is the y-intercept.
Rate of change
How much one quantity changes for each 1 unit of another. For a line, it is the slope.

Worked examples

Example 1. Read the slope and a point from each equation: (a) y − 6 = 4(x − 1), (b) y + 2 = −3(x + 5), (c) y = −(2/5)(x − 10).

  1. (a) The number multiplying the parenthesis is 4, so m = 4. The minus signs give positive coordinates: y1 = 6 and x1 = 1. The point is (1, 6).
  2. (b) The slope is −3 (the minus sign in front of the 3 belongs to the slope). Now flip the signs inside: y + 2 is y − (−2), so y1 = −2, and x + 5 is x − (−5), so x1 = −5. The point is (−5, −2).
  3. (c) The slope is −2/5. The letter y stands alone, so the left side is y − 0 and y1 = 0. Also x1 = 10. The point is (10, 0).
  4. Check (b): put in x = −5. Then y + 2 = −3(0) = 0, so y = −2, which gives (−5, −2). It matches.

Answer: Answer: (a) m = 4, point (1, 6). (b) m = −3, point (−5, −2). (c) m = −2/5, point (10, 0).

Example 2. Write the equation of the line with slope 5/2 that passes through (−6, 1).

  1. Start with the form: y − y1 = m(x − x1).
  2. Substitute m = 5/2, x1 = −6 and y1 = 1, with parentheses around the negative number: y − 1 = (5/2)(x − (−6)).
  3. Simplify the double negative: x − (−6) = x + 6. The equation is y − 1 = (5/2)(x + 6).
  4. Check the known point: at x = −6, the right side is (5/2)(0) = 0, so y = 1. It matches.
  5. Check another point: from (−6, 1) go right 2 and up 5 to (−4, 6). Then 6 − 1 = 5 and (5/2)(−4 + 6) = (5/2)(2) = 5. It matches.

Answer: Answer: y − 1 = (5/2)(x + 6)

Example 3. Write an equation of the line through (−4, 3) and (2, −1) in point-slope form, and then in slope-intercept form.

  1. Find the slope first: rise = −1 − 3 = −4 and run = 2 − (−4) = 6, so m = −4/6 = −2/3.
  2. Use (2, −1) as the known point: y − (−1) = −(2/3)(x − 2), which is y + 1 = −(2/3)(x − 2).
  3. Using (−4, 3) instead gives y − 3 = −(2/3)(x + 4). Both equations are correct.
  4. Convert the first one. Distribute: y + 1 = −(2/3)x + 4/3. Subtract 1: y = −(2/3)x + 4/3 − 3/3 = −(2/3)x + 1/3.
  5. Convert the second one. Distribute: y − 3 = −(2/3)x − 8/3. Add 3: y = −(2/3)x − 8/3 + 9/3 = −(2/3)x + 1/3. Both give the same line.

Answer: Answer: y + 1 = −(2/3)(x − 2) (or y − 3 = −(2/3)(x + 4)), which is y = −(2/3)x + 1/3.

Example 4. Write y + 1 = −(1/4)(x − 8) in slope-intercept form.

  1. Distribute −1/4 to both terms inside the parentheses: −(1/4)(x) = −(1/4)x and −(1/4)(−8) = +2. So y + 1 = −(1/4)x + 2.
  2. Subtract 1 from both sides: y = −(1/4)x + 1.
  3. So m = −1/4 and b = 1.
  4. Check with the known point (8, −1): −(1/4)(8) + 1 = −2 + 1 = −1. It matches.

Answer: Answer: y = −(1/4)x + 1

Example 5. Graph y − 2 = −(3/2)(x + 2) and say where it crosses the y-axis.

  1. Read the point: y − 2 gives y1 = 2, and x + 2 gives x1 = −2. Plot (−2, 2).
  2. Read the slope −3/2: a rise of −3 and a run of 2, which means down 3 and right 2.
  3. From (−2, 2) go right 2 and down 3 to (0, −1). Repeat to reach (2, −4). Going the other way, left 2 and up 3, reaches (−4, 5).
  4. Draw the line through the points and extend it in both directions.
  5. The line crosses the y-axis at (0, −1). Check by putting in x = 0: y − 2 = −(3/2)(2) = −3, so y = −1.

Answer: Answer: A line through (−4, 5), (−2, 2), (0, −1) and (2, −4). It crosses the y-axis at (0, −1).

Example 6. A water tank drains at a steady 10 liters per minute. After 5 minutes of draining it holds 150 liters. Let t be the minutes since draining started and V the liters in the tank. Write an equation, then find V after 12 minutes and the amount at the start.

  1. The slope is the rate of change. The tank loses 10 liters each minute, so m = −10 liters per minute.
  2. The known reading is 150 liters at 5 minutes: the point (5, 150).
  3. Write the equation: V − 150 = −10(t − 5).
  4. After 12 minutes: V − 150 = −10(12 − 5) = −70, so V = 80 liters.
  5. At the start, t = 0: V − 150 = −10(0 − 5) = 50, so V = 200 liters. In slope-intercept form, V = −10t + 200.
  6. The tank is empty when −10t + 200 = 0, at t = 20. So the equation fits the story only from t = 0 to t = 20, as long as the draining stays steady.

Answer: Answer: V − 150 = −10(t − 5), which is V = −10t + 200. After 12 minutes the tank holds 80 liters, and it started with 200 liters.

Example 7. Write an equation of the line through (2, −3) and (2, 5), and of the line through (−4, 6) and (7, 6).

  1. First pair: rise = 5 − (−3) = 8 and run = 2 − 2 = 0. The slope 8 ÷ 0 is undefined, so the line is vertical and has no point-slope equation. Every point has x = 2, so the equation is x = 2.
  2. Second pair: rise = 6 − 6 = 0 and run = 7 − (−4) = 11, so m = 0 ÷ 11 = 0.
  3. Use (−4, 6) as the known point: y − 6 = 0(x + 4). The right side is 0, so y − 6 = 0 and y = 6.

Answer: Answer: The first line is vertical: x = 2 (no point-slope form). The second is y − 6 = 0(x + 4), which is y = 6.

In plain English

Sometimes you know one spot on a line and how steep it is. You do not know where the line crosses the y-axis. Point-slope form is made for that.

The form is y − y1 = m(x − x1). The letter m is the slope. The pair (x1, y1) is the spot you know. The letters x and y stay as letters.

Here is the sneaky part. The form has minus signs built in. So x + 3 is really x − (−3), and the spot has an x-value of −3. Flip the sign you see.

If you only know two spots, find the slope first. Then use either spot. You will get two different-looking equations for one line, and that is fine.

See it

Point-slope form: what each part tells you y − y1 = m (x − x1 ) known y slope known x Example: y + 2 = −3(x + 5) y1 (the known y) y + 2 is y − (−2) so y1 = −2 m (the slope) the number multiplying the parenthesis so m = −3 x1 (the known x) x + 5 is x − (−5) so x1 = −5 The point is (−5, −2) and the slope is −3. Flip the sign next to x and y.
Reading y + 2 = −3(x + 5): the slope is the number in front of the parenthesis, and each point coordinate is the opposite of the sign you see.
−5−4−3−2−11234 54321−1−2−3 xy y = −(2/3)x + 1/3 (−4, 3) run 6 rise −4 (2, −1) 1. Find the slope firstrise = −1 − 3 = −4run = 2 − (−4) = 6m = −4/6 = −2/3 2. Use either pointUsing (2, −1):y + 1 = −(2/3)(x − 2)Using (−4, 3):y − 3 = −(2/3)(x + 4) Both: y = −(2/3)x + 1/3
From two points: find m first, then use either point. Both equations describe the same line, y = −(2/3)x + 1/3.

The same idea, the fun way

Pick the one that clicks. Each one explains the same concept from a different angle.

Sports

The mid-ride checkpoint

A cyclist starts a fitness timer partway along a route. When the timer shows 40 minutes, a sign says she has gone 11 kilometers along the route. She rides at a steady 0.25 kilometers per minute. At the 60-minute mark she has gone 11 + 0.25 × 20 = 16 kilometers.

How it connects: Let t be the minutes on the timer and d the kilometers along the route. The checkpoint is the point (40, 11) and the slope is 0.25, so d − 11 = 0.25(t − 40). The factor t − 40 is the time since the checkpoint, and 0.25(t − 40) is the distance gained since then. The limit: the equation only works while the pace stays the same. Real riders speed up and slow down.

Gaming

The glider’s descent

In a made-up game, you glide down toward the ground. At the 20-second mark your altitude is 300 meters, and you lose 15 meters every second. At the 30-second mark you are at 300 − 15(10) = 150 meters.

How it connects: Let x be the seconds and y the altitude in meters. The slope is −15 and the known point is (20, 300), so y − 300 = −15(x − 20). The minus sign in front of the 15 belongs to the slope (the altitude is falling). The signs inside the parentheses come only from the point. The limit: the formula stops fitting at ground level, where y = 0, which is at x = 40 seconds.

Idiom

Pick up where you left off

To “pick up where you left off” means to continue something from the exact place where you stopped.

How it connects: In point-slope form, (x1, y1) is where you left off, and the slope says how to continue from there: y − y1 = m(x − x1) starts at y1 and adds the slope times how far x has moved. You never have to walk back to the y-axis first. The limit: a line also extends backward from the known point, so the equation describes the points before it as well as the points after it.

Proverb

You can’t judge a book by its cover

This proverb says that how something looks on the outside does not tell you everything about what is inside.

How it connects: In x + 3, the plus sign makes it look like the point is about positive 3. Look inside: x + 3 = x − (−3), so x1 = −3. Rewrite it as a subtraction before you read the point. The limit: the cover is not random. In x + 3 or y − 5, the sign you see is the opposite of the coordinate’s sign, so one fixed rule (flip the sign) reads it correctly.

Nature

A cooling evening

On a clear evening the temperature is 18 degrees Celsius at 2 pm and 10 degrees Celsius at 6 pm. Use a 24-hour clock, so 2 pm is t = 14 and 6 pm is t = 18. At 8 pm (t = 20) the temperature is 6 degrees Celsius.

How it connects: Two readings give two points, (14, 18) and (18, 10), so find the slope first: (10 − 18) ÷ (18 − 14) = −2 degrees per hour. Then use either point: T − 18 = −2(t − 14). Using (18, 10) gives T − 10 = −2(t − 18), the same line. At t = 20 both give 6: 18 − 2(6) = 6 and 10 − 2(2) = 6. The limit: real temperatures do not fall at a perfectly steady rate all night, so the line is only a rough model for a few hours.

Everyday joke

The point and the slope

A point says, “I know where I am, but not where I’m going.” A slope says, “I know where I’m going, but not where I am.” They team up, and now there is a line.

How it connects: One point fits endless lines through it, and one slope fits endless parallel lines. A point plus a slope pins down exactly one line, which is why y − y1 = m(x − x1) needs both. The limit: this works when the slope is a number, so it leaves out vertical lines.

Memory help

Rhyme

Flip the sign

“Flip the sign by the x and the y, and the point is easy to spy.” (This is for the form y − y1 = m(x − x1).) For y + 4 = 3(x − 6), the y + 4 flips to y1 = −4, and the x − 6 flips to x1 = 6. The point is (6, −4). The slope 3 is not flipped: it keeps the sign in front of it.

Mnemonic

Slope, Pick, Plug

With two points, do three things in order: find the Slope, Pick one point, Plug into y − y1 = m(x − x1). Say “Slope, Pick, Plug.” If you skip the first step, there is no m to plug in.

Story

Since the checkpoint

Picture (x1, y1) as a checkpoint on a trail. The factor x − x1 is how far you have moved sideways since the checkpoint. Multiply by the slope m and you get how much you climbed (or dropped) since then. That is exactly y − y1. So the equation says: climb since the checkpoint = slope × sideways move since the checkpoint.

Picture it

A pin and an arrow

Draw a pin at (x1, y1) and a small staircase for the slope: across by the run, then up or down by the rise. The line goes through the pin along the staircase, in both directions.

Trap alert

The letters x and y stay

In y − 5 = 2(x − 3), do not put 3 in place of x or 5 in place of y. Those letters stand for every point on the line. Only the subscripted x1 and y1 get numbers. If you replace x and y too, you just get 0 = 0, which is true but is no longer an equation of a line.

Trap alert

The slope keeps its own sign

In y + 2 = −3(x + 5), the slope is −3. The plus signs inside make x1 and y1 negative, but they do not change the slope.

Study notes

Key ideas

  • Point-slope form is y − y1 = m(x − x1): m is the slope and (x1, y1) is any one known point on the line.
  • In y − y1 = m(x − x1), flip the sign you see next to x and y to read the point: x + 3 means x1 = −3, and y − 5 means y1 = 5. The slope keeps its own sign.
  • With two points, find the slope first, then use either point. Both choices describe the same line.
  • A non-vertical line has endless point-slope equations (one for each point on it) but only one slope-intercept equation.
  • To convert to slope-intercept form, distribute the slope and then move the number next to y to the right side.
  • A vertical line has an undefined slope, so it has no point-slope form. Its equation is x = a.

Steps

  1. If you have two points, find m = (y2 − y1) ÷ (x2 − x1). If the run is 0, stop: the line is vertical and its equation is x = a.
  2. Choose one point to be (x1, y1).
  3. Write y − y1 = m(x − x1), putting parentheses around negative numbers, then simplify the double negatives (y − (−2) is y + 2).
  4. If asked, distribute and solve for y to reach slope-intercept form.
  5. Check with the other point (or with the known point).

Formulas

  • Point-slope form: y − y1 = m(x − x1)
  • Slope from two points: m = (y2 − y1) ÷ (x2 − x1), assuming x2 ≠ x1
  • To slope-intercept form: y = mx + b, where b = y1 − m × x1
  • Horizontal line (m = 0): y − y1 = 0, which is y = y1
  • Vertical line: x = a (slope undefined, no point-slope form)

Watch out

  • Read the signs inside as opposites: y + 4 means y1 = −4, not 4.
  • Keep x and y as letters. Only x1, y1 and m are replaced by numbers.
  • With two points, find m first. Do not put both points into the form without a slope.
  • When you distribute, multiply the slope by both terms in the parentheses, including the sign of the second term.
  • A vertical line has an undefined slope, not a slope of 0.

Why it works

  • Why does the known point fit the equation? Put x = x1 in: the right side is m × 0 = 0, so y = y1.
  • Why do the signs flip? The form subtracts the known coordinate, and subtracting a negative number is the same as adding.
  • Why do all the point-slope equations of one line agree? Every pair of points on a line gives the same slope, so each equation says the same thing: rise = slope × run. After simplifying, they all give the same y = mx + b.

Common mistakes

✗ MistakeIn y + 3 = 2(x − 5), the point is (5, 3).
✓ InsteadThe y + 3 is y − (−3), so y1 = −3. The point is (5, −3). Flip the sign you see next to x and y.
✗ MistakeTo write the equation, replace x and y with the coordinates of the point.
✓ Insteadx and y stay as letters, because they stand for every point on the line. Only x1, y1 and m get numbers.
✗ MistakeEach line has exactly one point-slope equation.
✓ InsteadAny point on a non-vertical line can be the known point, so there are endless point-slope equations. They all simplify to the same y = mx + b.
✗ MistakeWith two points, put both into y − y1 = m(x − x1) and you are done.
✓ InsteadThe form needs m. Find the slope with (y2 − y1) ÷ (x2 − x1) first, then use only one point in the form.
✗ MistakeA vertical line has point-slope form with m = 0.
✓ Insteadm = 0 is a horizontal line. A vertical line has an undefined slope, so it has no point-slope form. Its equation is x = a.

Teach it back

Explain to a friend how to write the equation of the line through (−2, 9) and (4, 3) in point-slope form and in slope-intercept form, and why two different-looking point-slope equations can describe the same line.

Explaining it out loud shows what you really know. Did you…

  • I found the slope first: rise 3 − 9 = −6 and run 4 − (−2) = 6, so m = −1.
  • I picked a known point, for example (4, 3), and wrote y − 3 = −1(x − 4).
  • I distributed and added 3 to get y = −x + 7, then checked it with the other point: −(−2) + 7 = 9.
  • I explained that the other point (−2, 9) gives y − 9 = −1(x + 2), which simplifies to the same y = −x + 7.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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Practice it your way

Flashcards

Flip, recall and space your reviews.

Cheat sheet

One page. Everything you need.

True or false

Fast yes/no questions with reasons.

Matching game

Pair them up against the clock.

Practice questions

Step-up questions with explanations.

Slope of a LineSlope-Intercept FormGraphing Linear EquationsStandard Form of a LineParallel and Perpendicular Lines