How to Find Slope? (+FREE Worksheet!)
How to Find the Slope of a Line
Slope is one number that tells you how steep a line is and which way it tilts. Master it and you unlock graphing lines, writing equations in \(y=mx+b\), and every linear word problem you’ll meet. We’ll find slope from two points, from an equation, and straight off a graph, with a solver, drills, and a worksheet maker a tap away.
Find Slope: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Find slopeUse two points, a table, or the coefficient of x in slope-intercept form.
- Find an anchorUse a point or intercept so the line is in the right location.
- Check directionPositive slope rises left to right; negative slope falls left to right.
Worked examples
Find slope from two points
- Change in y is 10 – 4 = 6.
- Change in x is 3 – 1 = 2.
- Divide rise by run.
Write slope-intercept form
- Use y = mx + b.
- Put m = 3 and b = -2.
- Write the line.
Try one before moving on
Find Slope: pop-up practice
Picture walking up a hill. A gentle rise is easy; a steep one makes your legs burn. Slope is the math version of that feeling, a single number that captures how fast a line climbs or falls as you move to the right. Get comfortable with it and you’ll breeze through graphing lines, writing equations in \(y=mx+b\), and every linear word problem that comes your way. Let’s build it up slowly, one idea at a time.
What Is Slope?
Slope measures the tilt of a line as the ratio of how much it goes up or down to how much it goes across. We say rise over run. For any two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, the slope \(m\) is:
\(m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 – y_1}{x_2 – x_1}\)
How to find slope from two points (3 steps):
- Subtract the \(y\)-values for the rise.
- Subtract the \(x\)-values, in the same order, for the run.
- Divide and simplify. That’s your slope.
The sign tells you the direction and the size tells you the steepness. As a feel for the numbers: a slope of \(1\) is a 45° line, a slope of \(2\) is twice as steep, and a slope of \(\tfrac12\) is gentle.
Slope Comes in Three Disguises: Points, Equations, and Graphs
Slope shows up in three disguises, as two points, as an equation, and as a picture. Here’s how to handle each, and we’ll work all three below so you actually see them in action.
Use the formula
When you’re handed two coordinate pairs.
- Label them \((x_1,y_1)\) and \((x_2,y_2)\).
- Subtract the \(y\)’s for the rise.
- Subtract the \(x\)’s (same order!) for the run.
- Divide and simplify.
\(m=\dfrac{9-3}{4-1}=\dfrac{6}{3}=\) 2
Read off \(m\)
When the line is in slope-intercept form.
- Get the equation into \(y = mx + b\).
- The number multiplying \(x\) is the slope.
- The lone number \(b\) is the \(y\)-intercept.
Count rise over run
When you can see the line.
- Pick two points the line clearly crosses.
- Count up or down for the rise (up is +, down is −).
- Count across for the run (right is +, left is −).
- Write rise ÷ run.
Rise over run, right on the grid
This is the line through \((1,1)\) and \((3,5)\), the same points as Exercise 1 below. Going from the first point to the second, the line climbs up 4 (rise) and moves right 2 (run), so \(m=\tfrac{4}{2}=2\). Drop any two points into the solver to see the slope and the line appear.
Find a slopeWorked Examples
Read each one slowly and try to call out the next step before you see it. Each line is drawn so you can see the rise and run.
Example A, From two points
Find the slope through \((1,3)\) and \((4,9)\).
- Rise = difference of \(y\): \(9 – 3 = 6\).
- Run = difference of \(x\), same order: \(4 – 1 = 3\).
- Divide: \(m = \dfrac{6}{3} = 2\), up 2 for every step right.
Answer: \(m = 2\)
Example B, Negatives & a fraction
Find the slope through \((4,-6)\) and \((-3,-8)\).
- Rise: \(-8 – (-6) = -2\).
- Run, same order: \(-3 – 4 = -7\).
- Divide: \(m = \dfrac{-2}{-7} = \dfrac{2}{7}\). A clean fraction is the answer, don’t force a decimal.
Answer: \(m = \tfrac{2}{7}\)
Example C, From an equation
Find the slope of \(y = 3x + 6\).
- The equation is already in \(y = mx + b\) form.
- The slope is the coefficient of \(x\): \(m = 3\).
- The lone number, \(6\), is the \(y\)-intercept, the line crosses at \((0,6)\).
Answer: \(m = 3\)
Example D, From a graph
Use the line through \((1,1)\) and \((3,5)\).
- Pick the two marked points.
- Count from the first: up 4 (rise), right 2 (run).
- Divide: \(m = \dfrac{4}{2} = 2\).
Answer: \(m = 2\)
Example E, A vertical line (undefined)
Find the slope through \((4,1)\) and \((4,7)\).
- Rise: \(7 – 1 = 6\).
- Run: \(4 – 4 = 0\).
- \(m = \dfrac{6}{0}\), division by zero, so the slope is undefined. Every vertical line is like this.
Answer: undefined
Example F, A horizontal line (zero)
Find the slope through \((19,3)\) and \((20,3)\), the same flatness as the line drawn at \(y=3\).
- Rise: \(3 – 3 = 0\).
- Run: \(20 – 19 = 1\).
- \(m = \dfrac{0}{1} = 0\). Matching \(y\)-values always mean a flat line, slope zero.
Answer: \(m = 0\)
Example G, From a table
A table lists \(x = 1, 3\) with outputs \(y = 4, 10\). Find the slope.
- Read two rows as points: \((1,4)\) and \((3,10)\).
- Apply the formula: \(m = \dfrac{10 – 4}{3 – 1} = \dfrac{6}{2}\).
- Simplify: \(m = 3\). On a true line, any two rows give the same slope.
Answer: \(m = 3\)
Slope in the Wild
Slope isn’t just a classroom number. It’s the rate at which one thing changes against another. A car holding a steady 60 mph traces a distance-vs-time line with slope \(60\) (miles over hours). A wheelchair ramp that rises 4 inches over 12 inches of length has slope \(m=\tfrac{\text{rise}}{\text{run}}=\tfrac{4}{12}=\tfrac13\). A roof’s “pitch,” a hill’s grade, the cost per extra topping on a pizza, anywhere a quantity changes steadily, slope is the number measuring how fast.
Slip-Ups That Cost Easy Points
- Mixing up the order. If \((4,-6)\) is your first point, it stays first on top and bottom: \(\frac{-8-(-6)}{-3-4}\), never \(\frac{-8-(-6)}{4-(-3)}\). Pick an order and commit.
- Putting run over rise. Slope is rise over run, the \(y\)-difference goes on top. Flipping it is a classic.
- Subtracting negatives carelessly. Subtracting a negative adds: \(9-(-3)=12\). Write the double sign out so you don’t drop it.
- Confusing zero and undefined. A horizontal line has slope \(0\); a vertical line is undefined (division by zero). Flat is zero; straight up is undefined.
Your Turn: Find These Slopes
Work each one by hand, then reveal the answers. If one feels shaky, the step-by-step solver will walk you through it.
- \((1, 1)\) and \((3, 5)\)
- \((4, -6)\) and \((-3, -8)\)
- \((7, -12)\) and \((5, 10)\)
- \((19, 3)\) and \((20, 3)\)
- \((15, 8)\) and \((-17, 9)\)
- \((6, -12)\) and \((15, -3)\)
Show answers
- \(\color{blue}{2}\)
- \(\color{blue}{\frac{2}{7}}\)
- \(\color{blue}{-11}\)
- \(\color{blue}{0}\)
- \(\color{blue}{-\frac{1}{32}}\)
- \(\color{blue}{1}\)
Make Your Own Slope Worksheet
Want more reps? Generate a fresh worksheet with a full answer key, print it or save it as a PDF.
Frequently Asked Questions
What is a slope?
Slope is a number that tells you how steep a line is and which direction it tilts. It equals the rise (vertical change) divided by the run (horizontal change), written m = (y₂ − y₁)/(x₂ − x₁). A positive slope rises from left to right, a negative slope falls, a horizontal line has slope 0, and a vertical line has an undefined slope.
How do you find the slope from two points?
Subtract the y-values, subtract the x-values in the same order, and divide. For the points (2, −1) and (6, 7), the change in y is 7 − (−1) = 8 and the change in x is 6 − 2 = 4, so m = 8 ÷ 4 = 2. Keeping the order consistent is the key, because mixing it up flips the sign.
How do you find slope from a graph?
Pick two points where the line crosses grid corners exactly, then count the rise and the run between them. If you move up 3 squares and right 4 squares to get from one point to the other, the slope is 3/4. If you have to move down instead of up while going right, the slope is negative, such as −3/4.
How do you find the slope and y-intercept of y = −3x + 2?
When an equation is in slope-intercept form, y = mx + b, the number multiplying x is the slope and the constant is the y-intercept. For y = −3x + 2, the slope is −3 and the y-intercept is 2, so the line crosses the y-axis at (0, 2) and drops 3 units for every 1 unit you move right.
If the slope of a line is −5/2, how do you find the next point on the line?
Start at the point already graphed, move 5 units down and then 2 units to the right, and plot the new point. That works because −5/2 means a rise of −5 for a run of 2. You can also move 5 units up and 2 units left, since (+5)/(−2) is the same slope, −5/2.
How do you find slope from a word problem?
Treat the two situations in the problem as points (x, y) and find the rate of change between them. If a plumber charges $50 for a 1-hour job and $110 for a 3-hour job, the points are (1, 50) and (3, 110). The slope is (110 − 50) ÷ (3 − 1) = 30, meaning the cost goes up $30 for each extra hour.
Related Topics
Continue Your Study
Ready for the next step? Pick up right where this lesson leaves off:
Related to This Article
More math articles
- Top 10 HSPT Math Practice Questions
- Top 10 Pre-Algebra Prep Books (Our 2024 Favorite Picks)
- How to Select the Right Tool for the Job
- 10 Most Common 6th Grade PARCC Math Questions
- FREE 4th Grade MAP Math Practice Test
- Corporations, Trusts, and Vertical versus Horizontal Integration
- Full-Length 8th Grade Common Core Math Practice Test
- Algebra Puzzle – Challenge 39
- CPA BAR Forecasting: Read Assumptions Before the Model (2027 Prep)
- How to Compare and Order Rational Numbers?
What people say about "How to Find Slope from Two Points, a Graph or an Equation"?
No one replied yet.