➕ Algebra 1 · Linear Functions
Slope-Intercept Form
Read the slope and y-intercept straight from y = mx + b, write a line’s equation from a slope and a point or from two points, and explain m and b in real life.
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.
- 8.EE.B.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
- F-IF.C.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima.
- 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).
- F-LE.B.5: Interpret the parameters in a linear or exponential function in terms of a context.
By the end you can…
- Read the slope and y-intercept from an equation written as y = mx + b.
- Write the equation of a line from its slope and y-intercept, from its slope and a point, or from two points.
- Graph a line by plotting the y-intercept and using the slope to find more points.
- Explain what m and b mean in a real situation, and why a vertical line cannot be written in this form.
Lesson (school style)
Slope-intercept form is a very common way to write the equation of a line: y = mx + b. Two numbers describe any non-vertical line completely. The number m is the slope (how the line tilts), and the number b is the y-intercept (where the line crosses the y-axis). Once you can read them, you can sketch the line, write its equation, and explain what it means in a story.
The form y = mx + b
In y = mx + b, the letters x and y are the coordinates of any point on the line. The letters m and b stand for fixed numbers that belong to one particular line.
- m is the slope: the rise over run from the last lesson.
- b is the y-intercept: the y-value where the line crosses the y-axis. That point is (0, b), because every point on the y-axis has x = 0.
For example, the line y = 2x + 1 has slope 2 (up 2 for every 1 to the right) and crosses the y-axis at (0, 1).
Reading m and b from an equation
Find the number multiplied by x: that is m. Find the number added or subtracted on its own: that is b. Each number keeps the sign in front of it. Make sure the equation is solved for y, and put it in the order y = mx + b before you read it.
- If x has no number multiplying or dividing it, only a + or − sign, then m is 1 (as in y = x + 6) or −1 (as in y = −x + 6). If x is divided, as in x/4, that means (1/4)x, so m = 1/4.
- If there is no constant term, then b = 0 and the line passes through the origin (as in y = 3x).
- If there is no x term, then m = 0 and the line is horizontal (as in y = 9).
Graph it: start at b, then use m
To sketch a line from y = mx + b:
- Plot the y-intercept (0, b).
- Write the slope as rise over run. From your point, move the run to the right and the rise up (or down, if the rise is negative).
- Plot the new point, repeat once or twice, and draw a straight line through the points. Extend it in both directions: the line keeps going to the left of the y-axis too.
A slope of −3/4 means a rise of −3 and a run of 4: down 3, right 4. Keep the run going to the right and let the rise carry the sign.
A line with a slope other than 0 also crosses the x-axis, where y = 0. For y = (2/3)x − 4 that happens at (6, 0). These crossings are called intercepts. The line crosses the y-axis at (0, b), so the y-intercept is b (here −4), and it crosses the x-axis at (6, 0), so the x-intercept is 6.
Where does the formula come from?
Take a line with slope m that crosses the y-axis at (0, b). Pick any other point (x, y) on the line. Any two points on a line give the same slope (the rise-and-run triangles are similar, as in the last lesson), so the slope between (0, b) and (x, y) is m:
m = (y − b) ÷ (x − 0), which is m = (y − b) ÷ x (for x ≠ 0).
Multiply both sides by x to get mx = y − b. Add b to both sides: y = mx + b. So the formula is just the slope formula, rearranged. The point (0, b) fits too, because m(0) + b = b.
Writing an equation of a line
There are three common starting points.
- You know m and b. Fill them in. Slope 4 and y-intercept −5 give y = 4x − 5.
- You know the slope and one point (x1, y1). Put m into y = mx + b, substitute the point’s x and y, and solve for b.
- You know two points. Find the slope with the slope formula first, then do step 2 with either point. If the run is 0, stop: the line is vertical (see below).
m and b in real life
In a story, m is the rate of change: how much y changes for each 1 unit of x (the “per” number). And b is the starting value: the value of y when x = 0. Picture a membership with a monthly fee plus a price per visit. The fee is b (you pay it even with 0 visits), and the price per visit is m.
Include units: “3 dollars per visit,” not just “3.” Also remember that a story may use only part of the line. You cannot make −2 visits, so negative x values may make no sense.
The lines that do not fit
A vertical line, like x = 4, has an undefined slope (its run is 0). There is no number to use for m, so a vertical line cannot be written in slope-intercept form. We write it as x = a instead.
A horizontal line is fine: y = 5 is y = 0x + 5, with m = 0.
Many equations arrive in another form, like 3x + 6y = 18. If the equation has a y term, you can rearrange it into y = mx + b by solving for y, with the same steps you use on literal equations.
Key words
- 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.
- Slope
- How steep a line is and which way it tilts: rise divided by run. In y = mx + b, it is m.
- Y-intercept
- The y-value where a line crosses the y-axis. For y = mx + b it is b, and the line crosses at the point (0, b).
- Rate of change
- How much one quantity changes for each 1 unit of another. For a line, it is the slope.
- Starting value (initial value)
- The value of y when x = 0. In a story, it is the starting amount, and for a line it is b.
- Undefined slope
- The slope of a vertical line, where the run is 0 and division by 0 is not possible.
Worked examples
Example 1. Find the slope and y-intercept of each line: (a) y = 5x − 7, (b) y = −(1/3)x + 2, (c) y = 4 − 3x.
- (a) The number multiplying x is 5, so m = 5. The constant is −7 (the minus sign belongs to the 7), so b = −7. The line crosses the y-axis at (0, −7).
- (b) The number multiplying x is −1/3, so m = −1/3. The constant is 2, so b = 2.
- (c) The terms are in a different order. Rewrite it as y = −3x + 4. The sign stays with its number, so m = −3 and b = 4.
- Check (c): at x = 2, both 4 − 3(2) and −3(2) + 4 equal −2.
Answer: Answer: (a) m = 5, b = −7. (b) m = −1/3, b = 2. (c) m = −3, b = 4.
Example 2. Graph y = (2/3)x − 4.
- Read b = −4. Plot the y-intercept (0, −4).
- Read m = 2/3: rise 2, run 3. From (0, −4) go up 2 and right 3 to reach (3, −2).
- Repeat: up 2 and right 3 from (3, −2) to reach (6, 0), where the line crosses the x-axis.
- Draw a straight line through the points and extend it in both directions, past (0, −4) to the left too. Check: at x = 6, y = (2/3)(6) − 4 = 0. It matches.
Answer: Answer: A line through (0, −4), (3, −2) and (6, 0).
Example 3. Write the equation of the line with slope −2 that passes through (3, 1).
- Start with y = mx + b and put in m = −2: y = −2x + b.
- The point (3, 1) is on the line, so substitute x = 3 and y = 1: 1 = −2(3) + b.
- Simplify: 1 = −6 + b. Add 6 to both sides: b = 7.
- Write the equation: y = −2x + 7.
- Check the point: −2(3) + 7 = 1. It matches.
Answer: Answer: y = −2x + 7
Example 4. Write the equation of the line through (−2, −1) and (4, 8).
- Find the slope: rise = 8 − (−1) = 9 and run = 4 − (−2) = 6, so m = 9/6 = 3/2.
- Put m = 3/2 into y = mx + b and use the point (4, 8): 8 = (3/2)(4) + b.
- Simplify: 8 = 6 + b, so b = 2.
- The equation is y = (3/2)x + 2.
- Check with the other point (−2, −1): (3/2)(−2) + 2 = −3 + 2 = −1. It matches.
Answer: Answer: y = (3/2)x + 2
Example 5. A bike-share membership costs 6 dollars per month, and each ride costs 2 dollars. Write an equation for the monthly cost C in dollars after n rides. Say what m and b mean, then find the cost of 9 rides.
- The cost goes up 2 dollars for each ride, so the slope is m = 2 dollars per ride.
- With 0 rides you still pay the 6-dollar membership, so b = 6 dollars (the cost when n = 0).
- The equation is C = 2n + 6.
- For 9 rides: C = 2(9) + 6 = 18 + 6 = 24 dollars.
- Only n = 0, 1, 2, … makes sense here (you cannot take −3 rides), so only part of the line fits this story.
Answer: Answer: C = 2n + 6. The slope is 2 dollars per ride, the y-intercept is the 6-dollar monthly fee, and 9 rides cost 24 dollars.
Example 6. Write 3x + 6y = 18 in slope-intercept form, and name m and b.
- Solve for y, as you would for a literal equation. Subtract 3x from both sides: 6y = 18 − 3x.
- Divide every term by 6: y = 18/6 − 3x/6 = 3 − (1/2)x.
- Put the x term first: y = −(1/2)x + 3.
- So m = −1/2 and b = 3. Check with x = 2: y = −1 + 3 = 2, and 3(2) + 6(2) = 6 + 12 = 18. It matches.
Answer: Answer: y = −(1/2)x + 3, with m = −1/2 and b = 3
Example 7. Can each line be written in slope-intercept form? (a) The line through (1, 5) and (9, 5). (b) The line through (4, 1) and (4, 7).
- (a) Rise = 5 − 5 = 0 and run = 9 − 1 = 8, so m = 0 ÷ 8 = 0. The line crosses the y-axis at (0, 5), so b = 5. Yes: y = 0x + 5, which we write as y = 5.
- (b) Rise = 7 − 1 = 6 and run = 4 − 4 = 0, so the slope is undefined: 6 ÷ 0 is not a number. With no m, we cannot fill in y = mx + b.
- Every point on the line in (b) has the same x, so its equation is x = 4.
- So (a) fits the form, but (b) does not.
Answer: Answer: (a) Yes: y = 5 (slope 0). (b) No: the vertical line x = 4 has an undefined slope.
In plain English
A line is a straight path on a graph. To describe almost any line, you need two things: where it crosses the up-and-down axis, and how it moves.
Imagine a taxi that charges a starting fee the moment you get in, before the car moves at all. Then it charges a set price for every mile. The total is the starting fee plus the price per mile times the number of miles.
That is exactly what y = mx + b says. The number b is the starting amount. The number m is how much you add for each step in x. On a graph, b is where the line crosses the up-and-down y-axis, and m is how steep the line is.
One kind of line breaks the pattern: a straight-up wall. It has no slope number, so it does not fit y = mx + b. We write it as x equals a number.
See it
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Head start versus speed
Maya has 30 dollars saved and adds 5 dollars each week. Her friend Jordan has 10 dollars saved and adds 10 dollars each week. Maya starts ahead, but Jordan saves faster. After 4 weeks they both have 50 dollars, and after that Jordan stays ahead.
How it connects: Maya: y = 5x + 30. Jordan: y = 10x + 10. The y-intercept is the head start, and the slope is how fast you gain. Check week 4: 5(4) + 30 = 50 and 10(4) + 10 = 50. The limit: the lines only describe the story while each friend keeps saving the same amount.
Pizza by the topping
A pizza shop charges 9 dollars for a plain pizza and 1.50 dollars for each topping. The cost in dollars is y = 1.5x + 9, where x is the number of toppings. Four toppings cost 1.5(4) + 9 = 15 dollars.
How it connects: The y-intercept b = 9 is the price with 0 toppings (where x = 0), and the slope m = 1.5 is the rate of change: dollars per topping. The limit: you can only order whole toppings, so the real graph is a row of dots, not a full line.
Behind the starting line
A runner starts 18 meters behind the starting line and runs 6 meters per second. Measure her position from the line, counting positions past the line as positive. After x seconds her position is y = 6x − 18. She reaches the line when y = 0, which happens at x = 3 seconds.
How it connects: A negative y-intercept (b = −18) means you start on the negative side of the reference point, here 18 meters behind the line. The slope 6 is her speed in meters per second. Check: 6(3) − 18 = 0. The limit: the equation fits only from x = 0 while she keeps a steady 6 meters per second.
Walking to the treasure chest
In a made-up game, your character appears 36 steps from a treasure chest and walks toward it at 3 steps per second. The distance left after x seconds is y = −3x + 36.
How it connects: The slope −3 is negative because the distance shrinks as time moves right: 3 fewer steps each second. The y-intercept 36 is the starting distance. The limit: the formula only fits until the distance reaches 0, at x = 12 seconds. After that it would give negative distances, which make no sense for this story.
A journey of a thousand miles
The proverb says, “A journey of a thousand miles begins with a single step.” Big journeys start with one small move.
How it connects: Start drawing at (0, b). Every step of 1 to the right adds exactly m, the same amount for the first step as for the thousandth, because a line has a constant rate of change. The limit: real steps vary, but the steps along a line never do. And (0, b) is only where you start drawing: the line itself keeps going to the left of it too.
From scratch
To “start from scratch” means to begin with nothing already done or prepared.
How it connects: In a story, starting from scratch means a starting value of 0: b = 0, so the line passes through the origin (0, 0), like y = 9x. A story with b > 0 has a head start, and one with b < 0 starts below zero.
The vertical line at the door
A vertical line walks up to the Slope-Intercept Club. The doorkeeper says, “Sorry, members need an m.” The vertical line sighs, “My slope is undefined.” So it goes to the club next door, where every member’s name looks like x = 4.
How it connects: A vertical line has an undefined slope, so it has no number m and cannot be written as y = mx + b. Its equation is x = a. Horizontal lines are fine: y = b is the case m = 0.
Memory help
m for move, b for begin
m tells how the line moves (the slope). b tells where you begin drawing: the point (0, b) on the y-axis. Say it as: “Move with m, begin at b.” (The line itself keeps going both ways.)
Times the x
“Times the x, the slope you chart. All alone, that’s where you start.” In y = 4 − 3x, the number multiplying x is −3 (the slope), and the number all alone is 4 (where you start drawing). Careful: x/4 means (1/4)x, so its slope is 1/4, not 4.
Read it like a receipt
Read y = mx + b as a sentence: “Start with b, then add m for every 1 of x.” For y = 6x + 3, that says: start with 3, then add 6 for every 1 of x. If m is negative, “add m” really means you lose that much.
Draw the staircase
Plot (0, b) first. Then draw a staircase from that point: across by the run, then up or down by the rise. Each stair corner is another point on the line. Label the stairs before you plot.
The sign stays with its number
In y = 4 − 3x, the slope is −3, not 3, and b is 4. In y = 5x − 7, b is −7. Rewrite the equation in the order y = mx + b before you read it.
Vertical lines are not members
x = a cannot be written as y = mx + b, because its slope is undefined. Do not try m = 0: that makes a horizontal line, which is a different line.
Study notes
Key ideas
- Slope-intercept form is y = mx + b: m is the slope and b is the y-intercept, so the line crosses the y-axis at (0, b).
- In a story, m is the rate of change (the “per” number) and b is the starting value, the value of y when x = 0.
- To write an equation from a slope and a point, or from two points, find m first, then substitute a point to solve for b.
- A vertical line cannot be written in slope-intercept form. It is x = a, and its slope is undefined.
- An equation with a y term, like 3x + 6y = 18, can be rearranged into slope-intercept form by solving for y.
Steps
- If you have two points, find the slope: m = (y2 − y1) ÷ (x2 − x1). If the run is 0, stop: the line is vertical and its equation is x = a.
- Write y = mx + b with your m in place.
- Substitute the x and y of one point and solve for b.
- Write the final equation and check it with the other point.
Formulas
- Slope-intercept form: y = mx + b
- Finding b from a point: b = y1 − m × x1, using any point (x1, y1) on the line
- Horizontal line: y = b, slope 0
- Vertical line: x = a, slope undefined, not in slope-intercept form
Watch out
- Signs belong to their numbers: in y = 5x − 7, b = −7, not 7.
- Put the equation in the order y = mx + b first. In y = 4 − 3x, the slope is −3.
- Do not swap them: m goes with x, and b is alone. In y = 4x + 9, the slope is 4 and the y-intercept is 9, so the line crosses the y-axis at (0, 9).
- When you divide to solve for y, divide every term.
- A point’s y-value is not b unless its x is 0.
Why it works
- Why does b sit at (0, b)? Setting x = 0 makes mx = 0, which leaves y = b.
- Why is y = mx + b the slope formula in disguise? Use the points (0, b) and (x, y) in m = (y − b) ÷ x, then solve for y.
- Why can’t a vertical line fit? Its run is 0, so its slope is undefined: there is no number to put in for m.
Common mistakes
Teach it back
Explain to a friend how to write the equation of the line through (1, 2) and (4, 11), and what m and b tell you about the line.
Explaining it out loud shows what you really know. Did you…
- I found the slope: rise 11 − 2 = 9 over run 4 − 1 = 3, so m = 3.
- I put m and one point into y = mx + b and solved for b to get b = −1.
- I wrote y = 3x − 1 and checked it with the other point.
- I said what m and b mean: up 3 for each 1 step right, and the line crosses the y-axis at (0, −1).
Your 3-week memory plan
- TodayRead the lesson, then try the flashcards.
- TomorrowRecall it from memory first, then check.
- In 3 daysTake the true/false and practice questions.
- In 1 weekMix it with older topics and teach it back.
- In 3 weeksA last quick review before it fades.
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