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Finding Slope from a Graph, Table or Equation

Find and compare the slope of a line from a graph, two points, a table or an equation.

Practice this skill 20 min Slope from a graph, equation or table Leer en español
Key idea

What you'll learn

Slope is the single most important number about a line. It tells you how steep the line is and which way it goes.

In this lesson you will find slope three ways:

  • from a graph (count rise and run);
  • from two points or a table (use the slope formula);
  • from an equation (solve for y and read m).

You will also learn the four kinds of slope (positive, negative, zero, undefined), how to compare steepness, and how to work backwards to a missing coordinate.

Why it matters on the GED. Slope appears in several questions on almost every Math test: "What is the slope of the line shown?", "Which equation has the greatest slope?", "Find the missing value if the slope is …". It is also the foundation for unit rates, graphing lines and writing equations — all later lessons build on it.

Real life. The pitch of a roof, the grade of a road or wheelchair ramp, the price per gallon on a receipt, and the speed of a car on a distance–time graph are all slopes.

Words to know

  • slope (m) — the rate a line goes up or down: change in y divided by change in x (a slope of 2 means up 2 for every 1 to the right).
  • rise — the vertical change between two points (from y = 1 to y = 7, the rise is 6).
  • run — the horizontal change between two points (from x = 2 to x = 5, the run is 3).
  • positive slope — the line goes up from left to right (e.g. m = 3).
  • negative slope — the line goes down from left to right (e.g. m = -\tfrac{1}{2}).
  • zero slope — a horizontal line such as y = 4.
  • undefined slope — a vertical line such as x = -2 (run is 0, and we cannot divide by 0).
  • slope-intercept form — y = mx + b; m is the slope, b is the y-intercept (e.g. y = 5x - 1 has m = 5).
  • standard form — Ax + By = C (e.g. 4x + 2y = 10); must be rearranged to see the slope.
  • y-intercept — where a line crosses the y-axis (in y = 5x - 1 it is (0, -1)).
  • steepness — the size of the slope ignoring its sign, \lvert m \rvert.

All Math words in the glossary

Slope from a graph: rise over run

Coordinate plane from −5 to 5. A line passes through the points (−3, −1) and (3, 3). A dashed right triangle shows the run of 6 units to the right from (−3, −1) to (3, −1) and the rise of 4 units up from (3, −1) to (3, 3). Slope = 4/6 = 2/3.−5−4−3−2−112345−5−4−3−2−1123450xyrun = 6rise = 4(−3, −1)(3, 3)

Slope compares the up/down change to the left/right change:

m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x}

Steps for a graph:

  1. Pick two points where the line passes exactly through grid corners.
  2. Move from the left point to the right point. Count the rise (up is +, down is -).
  3. Count the run (to the right, always + when you go left to right).
  4. Divide rise by run and simplify.

Example. The line below passes through (-3, -1) and (3, 3). Rise = 3 - (-1) = 4; run = 3 - (-3) = 6. So m = \frac{4}{6} = \frac{2}{3}.

Check the sign with your eyes. Read the graph left to right like a sentence. Uphill → positive. Downhill → negative.

Check the scale. If each grid square is 5 units on the y-axis and 1 unit on the x-axis, counting squares is not enough — use the axis labels to find the real rise and run.

The four kinds of slope

Four lines on a coordinate plane: a line rising from left to right labeled positive slope, a line falling from left to right labeled negative slope, a horizontal line labeled zero slope, and a vertical line labeled undefined slope.−5−4−3−2−112345−5−4−3−2−1123450xypositivenegativezeroundefined
Line looks likeSlopeExampleWhy
rising ↗positivey = 2x + 1y grows as x grows
falling ↘negativey = -3x + 4y shrinks as x grows
flat —zeroy = 4rise = 0, so \frac{0}{\text{run}} = 0
straight up and down |undefinedx = -2run = 0, and \frac{\text{rise}}{0} is not a number
Four small graphs comparing the kinds of slope. Top left: y = 2x + 1 rises from left to right; from (0, 1) the run is 1 and the rise is 2, so the slope is positive, 2. Top right: y = −3x + 4 falls; from (0, 4) the run is 1 and the rise is −3, so the slope is negative, −3. Bottom left: y = 4 is horizontal; the rise is 0, so the slope is zero. Bottom right: x = −2 is vertical; the run is 0, so the slope is undefined.Positive slopexy = 2x + 1Negative slopexy = −3x + 4Zero slopexy = 4Undefined slopexx = −2rise 2run 1m = 2run 1rise −3m = −3rise = 0m = 0run = 0m = rise/0undefined

Steepness. A bigger \lvert m \rvert means a steeper line. m = -4 is steeper than m = 3 because \lvert -4 \rvert = 4 > 3. A slope between -1 and 1 (like \frac{1}{4}) is gentle.

Three lines through the origin compared for steepness. Left: slope 3, run 1 and rise 3. Middle: slope −4, run 1 and rise −4; it is the steepest because |−4| = 4 is the largest size. Right: slope 1/4, run 4 and rise 1; it is gentle because 1/4 is between −1 and 1.run 1rise 3m = 3|m| = 3run 1rise −4m = −4|m| = 4 steepestrun 4rise 1m = 1/4|m| = 1/4 gentle

Careful with words: "greatest slope" means the largest signed number (3 > -4), but "steepest" means the largest size (\lvert -4 \rvert > \lvert 3 \rvert).

Slope from two points, a table, or an equation

Two points. Use the slope formula with (x_1, y_1) and (x_2, y_2):

m = \frac{y_2 - y_1}{x_2 - x_1}

Example: (-3, 5) and (1, -3) → m = \frac{-3 - 5}{1 - (-3)} = \frac{-8}{4} = -2. You may pick either point as "point 1" — just use the same order on top and bottom.

Coordinate plane from −5 to 3 by −5 to 6. A falling line passes through (−3, 5) and (1, −3). A dashed right triangle shows the run of 1 − (−3) = 4 to the right and the rise of −3 − 5 = −8 (8 down). Slope = −8/4 = −2.−5−4−2−123−5−4−2−1123460xyrun = 4rise = −8(−3, 5)(1, −3)m = −8/4 = −2

A table. Choose any two rows (or columns) and use the formula. If x goes up by 2 and y goes up by 6 each time, the slope is \frac{6}{2} = 3, not 6.

x246
y111723

m = \frac{17 - 11}{4 - 2} = \frac{6}{2} = 3.

Graph of the table x = 2, 4, 6 and y = 11, 17, 23. The points (2, 11), (4, 17) and (6, 23) lie on a line. Between neighbouring points the run is 2 and the rise is 6, so the slope is 6/2 = 3, not 6.123456748121620240xyrun 2rise 6run 2rise 6(2, 11)(4, 17)(6, 23)m = 6/2 = 3

If the change in y is not the same for equal changes in x, the table is not a line, and it has no single slope.

An equation. Solve for y to get y = mx + b; then m is the coefficient of x.

  • y = -3x + 7 → m = -3 (already solved).
  • 4x + 2y = 10 → 2y = -4x + 10 → y = -2x + 5 → m = -2.
  • y = 6 - \frac{1}{2}x → reorder: y = -\frac{1}{2}x + 6 → m = -\frac{1}{2}.
  • Shortcut for Ax + By = C: m = -\frac{A}{B}.

Work backwards. If you know the slope and one coordinate is missing, put a letter in the formula and solve. Example: slope \frac{5}{2} through (2, k) and (6, 13) → \frac{13 - k}{4} = \frac{5}{2} → 13 - k = 10 → k = 3.

First-quadrant grid from 0 to 7 by 0 to 14. A line with slope 5/2 passes through (2, k) and (6, 13). The run from x = 2 to x = 6 is 4, so the rise must be 5/2 × 4 = 10, and k = 13 − 10 = 3. The lower point is (2, 3).123456724681012140xyrun = 4rise = 10(2, k)(6, 13)10/4 = 5/2k = 13 − 10 = 3
Formula

Formulas (on the GED formula sheet)

m = \frac{y_2 - y_1}{x_2 - x_1}
  • m = slope
  • (x_1, y_1) and (x_2, y_2) = any two points on the line
y = mx + b
  • m = slope, b = y-intercept (the y-value where x = 0)

Both formulas are on the GED formula sheet (listed under slope and slope-intercept form), so you don't need to memorize them — but you do need to know how to use them quickly. Not on the sheet (memorize): horizontal lines have slope 0, vertical lines have undefined slope, and the shortcut m = -\frac{A}{B} for Ax + By = C.

Worked example

Example 1 — From two points

Problem. Find the slope of the line through (2, -1) and (6, 7).

  1. y_2 - y_1 = 7 - (-1) = 8.
  2. x_2 - x_1 = 6 - 2 = 4.
  3. m = \frac{8}{4} = 2.
Coordinate plane from −1 to 7 by −2 to 8. The line through (2, −1) and (6, 7). A dashed right triangle shows run 6 − 2 = 4 and rise 7 − (−1) = 8, so the slope is 8/4 = 2. Green steps show right 1, up 2 four times through (3, 1), (4, 3) and (5, 5).−112457−212345680xyrun = 4rise = 8steps: right 1, up 2(2, −1)(6, 7)m = 8/4 = 2

Answer: m = 2.

Check: From (2, -1), go right 1 and up 2 four times: (3, 1), (4, 3), (5, 5), (6, 7) ✓

Worked example

Example 2 — From an equation in standard form

Problem. What is the slope of 5x - 3y = 12?

  1. Subtract 5x: -3y = -5x + 12.
  2. Divide every term by -3: y = \frac{5}{3}x - 4.
  3. Read the coefficient of x: m = \frac{5}{3}.
Graph of 5x − 3y = 12, rewritten as y = (5/3)x − 4, on a plane from −2 to 7 by −6 to 4. The line crosses the y-axis at (0, −4). A dashed right triangle from (0, −4) runs 3 right and rises 5 up to (3, 1), showing the slope 5/3.−2−1124567−6−5−3−2−12340xyrun = 3rise = 5(0, −4)(3, 1)y = (5/3)x − 4m = 5/3

Answer: m = \tfrac{5}{3}.

Check: Shortcut -\frac{A}{B} = -\frac{5}{-3} = \frac{5}{3} ✓

Worked example

Example 3 — Real world: a wheelchair ramp

Problem. Building code says a wheelchair ramp may have a slope of at most \frac{1}{12}. A ramp rises 2 feet over a horizontal distance of 20 feet. Does it meet the code?

  1. Slope = \frac{\text{rise}}{\text{run}} = \frac{2}{20} = \frac{1}{10} = 0.1.
  2. The limit is \frac{1}{12} \approx 0.083.
  3. Compare: 0.1 > 0.083, so the ramp is too steep.
Side view of two ramps drawn to scale, each rising 2 feet to the same landing. The real ramp (blue) has a run of 20 feet, so its slope is 2/20 = 1/10, which is too steep. The code-limit ramp (green, dashed) needs a run of 24 feet, so its slope is 2/24 = 1/12, which just meets the code.landingrise 2 ftreal ramp: run 20 ft → slope 2/20 = 1/10 (too steep)code limit: run 24 ft → slope 2/24 = 1/12 (OK)Side view, drawn to scale

Answer: No — its slope \frac{1}{10} is greater than \frac{1}{12}. (A 2-ft rise needs a run of at least 2 \times 12 = 24 ft.)

Check: \frac{2}{24} = \frac{1}{12} exactly meets the limit ✓

Worked example

Example 4 — Real world: a table of costs

Problem. A plumber's bill depends on hours worked.

Hours235
Bill ($)190255385

What is the slope, and what does it mean?

  1. Use the first two columns: \frac{255 - 190}{3 - 2} = \frac{65}{1} = 65.
  2. Confirm with the last two: \frac{385 - 255}{5 - 3} = \frac{130}{2} = 65 ✓ (same slope, so it is linear).
Graph of a plumber's bill against hours worked, with points (2, 190), (3, 255) and (5, 385) on a straight line. From 2 to 3 hours the run is 1 and the rise is 65 dollars; from 3 to 5 hours the run is 2 and the rise is 130 dollars. Both give a slope of 65 dollars per hour.123456501001502002503003504004500Hours workedBill ($)1652130(2, 190)(3, 255)(5, 385)65/1 = 130/2 = $65 per hour

Answer: m = 65 — the bill rises $65 per hour of work.

Check: 190 + 65 = 255 and 255 + 2(65) = 385 ✓

Worked example

Example 5 — GED-hard: work backwards

Problem. A line has slope -\frac{3}{4} and passes through (-2, 5) and (x, -1). Find x.

  1. Rise from y = 5 to y = -1: -1 - 5 = -6.
  2. Slope = rise/run, so -\frac{3}{4} = \frac{-6}{\text{run}}.
  3. Solve: \text{run} = \frac{-6}{-3/4} = -6 \times \left(-\frac{4}{3}\right) = 8.
  4. x = -2 + 8 = 6.
Coordinate plane from −3 to 7 by −2 to 6. A line with slope −3/4 passes through (−2, 5) and (6, −1). A dashed right triangle shows the rise of −1 − 5 = −6 (6 down) and the run of 8, so x = −2 + 8 = 6.−3−2−1123457−2−1123460xyrun = 8rise = −6(−2, 5)(x, −1) = (6, −1)−6/8 = −3/4

Answer: x = 6.

Check: \frac{-1 - 5}{6 - (-2)} = \frac{-6}{8} = -\frac{3}{4} ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

  • Enter the whole slope as one fraction with the n/d key: press n/d, type the top (e.g. 7 − (−) 1), press the down arrow, type the bottom (6 − 2), then enter. The calculator shows 2.
  • Use the (−) key for negative coordinates, not the minus key. 1 - (-3) is 1 − (−) 3.
  • Press toggle to switch an answer like \frac{3}{50} to 0.06 (or back), so you can match the answer choices.
  • When NOT to use it: reading rise and run off a graph, or reading m from an equation already in y = mx + b form — both are faster by eye. Small-number slope questions are often doable in your head.
Common mistake

Common mistakes

  • Run over rise. Writing \frac{\text{change in } x}{\text{change in } y} gives \frac{2}{3} instead of \frac{3}{2}. ✗ y (rise) always goes on top. ✓
  • Mixing the order. Using y_2 - y_1 on top but x_1 - x_2 on the bottom flips the sign. ✗ Start both subtractions with the same point. ✓
  • Sign errors with negatives. 1 - (-3) is 4, not -2. ✗ Put negatives in parentheses and change "minus a negative" to "plus." ✓
  • Reading the slope before solving for y. In 4x + 2y = 10 the slope is not 4. ✗ Solve: y = -2x + 5, so m = -2. ✓
  • Confusing m and b. In y = -3x + 7, 7 is the y-intercept, not the slope. ✗ The slope is the number attached to x. ✓
  • Mixing up zero and undefined. Horizontal lines (y = 5) have slope 0; vertical lines (x = -2) have undefined slope. ✓
Test strategy

GED test strategy

  • Sign first. Before computing, decide whether the line rises or falls. Eliminate every answer with the wrong sign — often half the choices.
  • Expect the trap answers. Choices usually include the reciprocal (\frac{2}{3} vs \frac{3}{2}), the opposite sign, and the y-intercept. Double-check which one you computed.
  • Use the formula sheet. The slope formula and y = mx + b are printed there — glance at it rather than trusting memory under stress.
  • Plug back in. For "find the missing coordinate," put your answer back into the slope formula to see that it gives the stated slope.
  • Simplify before comparing. \frac{-4}{6} and -\frac{2}{3} are the same slope; answer choices are usually in lowest terms.
  • Read the question twice: "steepest" vs "greatest slope" vs "least slope" are different questions.

Check yourself

Slope from a graph, equation or table

What is the slope of the line that passes through the points (1,\ 2) and (4,\ 8)?

Answer choices
Hint

Slope is the change in y divided by the change in x. Keep the points in the same order on top and bottom.

Check yourself

Slope from a graph, equation or tableNo calculator

What is the slope of the line y = -3x + 7?

Answer choices
Hint

This equation is already in the form y = mx + b. Which letter is the slope?

Summary

  • Slope m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1} (on the formula sheet).
  • Rising = positive, falling = negative, horizontal = 0, vertical = undefined.
  • From a graph: pick two exact grid points, count rise and run, check the scale.
  • From a table: divide the change in y by the change in x — not just the y-jump.
  • From an equation: solve for y; the slope is the coefficient of x (shortcut -\frac{A}{B} for Ax + By = C).
  • Steepness compares \lvert m \rvert: a slope of -4 is steeper than 3.
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