GED Free
Español Get the app
Menu

Graphing Linear Equations

Graph lines from slope-intercept form, intercepts or a table, and write the equation of a line from its graph.

Practice this skill 22 min Graph linear equations Leer en español
Key idea

What you'll learn

In this lesson you will learn to draw the graph of a linear equation — and to go the other way, from a graph to its equation.

You will be able to:

  • graph y = mx + b using the y-intercept and slope;
  • graph Ax + By = C using the two intercepts;
  • graph from a table of values;
  • decide whether a point is on a line;
  • write the equation of a line from its graph or from two points;
  • explain what the slope and intercept mean in a real situation.

Why it matters on the GED. You may be asked to click a point on a line, choose which graph matches an equation, choose which equation matches a graph, or explain what the starting value means in a cost graph. The GED formula sheet gives you y = mx + b and point-slope form — this lesson shows you how to use them.

Real life. Phone plans, taxi fares, gym memberships and hourly jobs with a sign-up fee all have a starting amount plus a rate — a straight-line graph.

Words to know

  • linear equation — an equation whose graph is a straight line (e.g. y = 2x - 1, 3x + 4y = 24).
  • slope-intercept form — y = mx + b, with slope m and y-intercept b (y = 4x - 5: m = 4, b = -5).
  • standard form — Ax + By = C (3x + 4y = 24).
  • point-slope form — y - y_1 = m(x - x_1), for a line with slope m through (x_1, y_1).
  • y-intercept — where the line crosses the y-axis; x = 0 there (e.g. (0, -5)).
  • x-intercept — where the line crosses the x-axis; y = 0 there (e.g. (3, 0)).
  • solution of an equation — a point (x, y) that makes the equation true; every solution lies on the graph.
  • table of values — a list of x-values and the y-values the equation gives for them.
  • initial value — the y-intercept in a real-world problem (a fee or starting amount).

All Math words in the glossary

Method 1: slope and y-intercept

Graph of y = (2/3)x − 1 on a coordinate plane from −5 to 5. The line starts at the y-intercept (0, −1); a dashed step goes right 3 and up 2 to the point (3, 1). The line also passes through (−3, −3).−5−4−3−2−112345−5−4−3−2−1123450xyright 3up 2start (0, −1)(3, 1)

When the equation is in the form y = mx + b:

  1. Plot the y-intercept (0,\ b) on the y-axis.
  2. Use the slope m = \frac{\text{rise}}{\text{run}} to move to a second point. Rise up if positive, down if negative; run to the right.
  3. Repeat once more to get a third point (a built-in check).
  4. Draw a straight line through the points, across the whole grid.

Example. Graph y = \frac{2}{3}x - 1.

  • Start at (0, -1).
  • Slope \frac{2}{3}: up 2, right 3 → (3, 1).
  • Going the other way (down 2, left 3) → (-3, -3).

Whole-number slopes. Write them over 1: m = -3 = \frac{-3}{1} → down 3, right 1.

Negative fraction slopes. m = -\frac{2}{5} → down 2, right 5 (or up 2, left 5 — same line).

Missing b. y = 4x means b = 0: start at the origin. Missing x. y = 3 is a horizontal line through (0, 3). Missing y. x = -2 is a vertical line through (-2, 0).

Coordinate plane from −5 to 5 with three special lines. y = 4x (blue) has b = 0, so it starts at the origin and rises 4 for every 1 to the right. y = 3 (green) is a horizontal line through (0, 3). x = −2 (red) is a vertical line through (−2, 0).−5−4−12345−5−41250xy41(0, 3)(−2, 0)y = 4xy = 3x = −2

If the equation is not solved for y, solve it first: 4x + 2y = 10 \to y = -2x + 5.

Method 2: the two intercepts

Graph of 2x + 3y = 6 on a coordinate plane from −5 to 5. The line crosses the x-axis at the x-intercept (3, 0) and the y-axis at the y-intercept (0, 2).−5−4−3−2−112345−5−4−3−2−1123450xyx-intercept (3, 0)y-intercept (0, 2)

When the equation is in standard form Ax + By = C, the intercepts are usually fastest.

  1. x-intercept: set y = 0 and solve for x.
  2. y-intercept: set x = 0 and solve for y.
  3. Plot both and draw the line through them.

Example. Graph 2x + 3y = 6.

  • y = 0: 2x = 6, so x = 3 → (3, 0).
  • x = 0: 3y = 6, so y = 2 → (0, 2).
  • Draw the line through (3, 0) and (0, 2). (Its slope is \frac{0 - 2}{3 - 0} = -\frac{2}{3}.)

This is called the cover-up method: cover the y-term and solve what's left for x; cover the x-term and solve for y.

Watch the signs. For 5x - 2y = 10: x = 0 gives -2y = 10, so y = -5 (not +5).

Graph of 5x − 2y = 10 on a plane from −3 to 5 by −6 to 4. Covering the y-term gives 5x = 10, so the x-intercept is (2, 0). Covering the x-term gives −2y = 10, so the y-intercept is (0, −5), below the x-axis, not (0, 5). The line through them rises with slope 5/2.−3−2−11345−6−4−3−2−112340xy(2, 0)(0, −5)Cover y (y = 0): 5x = 10 → x = 2Cover x (x = 0): −2y = 10 → y = −5

Method 3: a table of values. Choose 3 easy x-values (like -2, 0, 2), compute y for each, plot, and connect. For fractional slopes, pick x-values that are multiples of the denominator so y comes out whole.

x-303
y = \tfrac{2}{3}x - 1-3-11

If your three points don't line up, you made an arithmetic mistake.

A table of values for y = (2/3)x − 1 with x = −3, 0, 3 gives y = −3, −1, 1. The three points (−3, −3), (0, −1) and (3, 1) are plotted and lie on one straight line, which checks the arithmetic.x−303y−3−11y = (2/3)x − 1x = multiples of 3so y comes out whole−5−4−2−11245−5−4−223450xy(−3, −3)(0, −1)(3, 1)

From a graph (or two points) to an equation

From a graph.

  1. Read b where the line crosses the y-axis.
  2. Find the slope from two clear grid points.
  3. Write y = mx + b.

Example: a line crosses at (0, 2) and passes through (4, -1) → b = 2, m = \frac{-1 - 2}{4 - 0} = -\frac{3}{4} → y = -\frac{3}{4}x + 2.

A line crosses the y-axis at (0, 2), so b = 2. It also passes through (4, −1). A dashed right triangle from (0, 2) runs 4 right and drops 3, so m = −3/4. The equation is y = −(3/4)x + 2.−3−2−112356−4−3−213450xyrun = 4rise = −3b = 2(4, −1)y = −(3/4)x + 2

From two points (no y-intercept given).

  1. Find m with the slope formula.
  2. Substitute m and one point into y = mx + b, solve for b.

Example: (-2, 7) and (4, -5) → m = \frac{-12}{6} = -2. Then 7 = -2(-2) + b \Rightarrow b = 3 → y = -2x + 3.

Coordinate plane from −3 to 5 by −6 to 8. A line passes through (−2, 7) and (4, −5). A dashed right triangle shows run 6 and rise −12, so m = −12/6 = −2. Substituting gives b = 3, and the line crosses the y-axis at (0, 3): y = −2x + 3.−3−11235−6−4−3−2−11245680xyrun = 6rise = −12(−2, 7)(4, −5)(0, 3): b = 3m = −12/6 = −2 → y = −2x + 3

Is a point on the line? Substitute its x and y. If the equation is true, the point is on the graph; if not, it isn't. (2, 4) on y = 3x - 2? 3(2) - 2 = 4 ✓

Real-world meaning.

  • b = starting value (sign-up fee, initial balance, flat charge).
  • m = rate of change (per month, per mile, per hour).

A gym that charges $50 to join plus $25 a month: C = 25m + 50.

Formula

Formulas (on the GED formula sheet)

y = mx + b
  • m = slope, b = y-intercept, (x, y) = any point on the line.
y - y_1 = m(x - x_1)
  • point-slope form: m = slope, (x_1, y_1) = a known point on the line.
m = \frac{y_2 - y_1}{x_2 - x_1}

All three are on the GED formula sheet. Memorize these extra rules (not on the sheet): x-intercept → set y = 0; y-intercept → set x = 0; y = k is horizontal; x = k is vertical.

Worked example

Example 1 — Graph using slope and intercept

Problem. Graph y = -3x + 2. Name two points the line passes through besides the y-intercept.

  1. y-intercept: (0, 2).
  2. Slope -3 = \frac{-3}{1}: down 3, right 1 → (1, -1).
  3. Again: down 3, right 1 → (2, -4).
Graph of y = −3x + 2 on a plane from −3 to 4 by −6 to 5. It starts at the y-intercept (0, 2). Two dashed steps each go right 1 and down 3, reaching (1, −1) and then (2, −4).−3−2−134−6−5−3−213450xy1−31−3(0, 2)(1, −1)(2, −4)y = −3x + 2down 3, right 1

Answer: e.g. (1, -1) and (2, -4).

Check: -3(2) + 2 = -4 ✓

Worked example

Example 2 — Graph using intercepts

Problem. Find the intercepts of 4x - 3y = 12 and describe the graph.

  1. y = 0: 4x = 12 \Rightarrow x = 3 → (3, 0).
  2. x = 0: -3y = 12 \Rightarrow y = -4 → (0, -4).
  3. Slope = \frac{0 - (-4)}{3 - 0} = \frac{4}{3}, so the line rises.
Graph of 4x − 3y = 12 on a plane from −3 to 6 by −6 to 3. The x-intercept is (3, 0) and the y-intercept is (0, −4). A dashed right triangle from (0, −4) runs 3 right and rises 4 up to (3, 0), so the slope is 4/3 and the line rises.−3−2−112456−6−5−3−2−11230xyrun = 3rise = 4(3, 0)(0, −4)4x − 3y = 12m = 4/3

Answer: intercepts (3, 0) and (0, -4); a rising line.

Check: shortcut -\frac{A}{B} = -\frac{4}{-3} = \frac{4}{3} ✓

Worked example

Example 3 — Real world: taxi fare

Problem. A taxi charges a flat $3.50 plus $2.25 per mile. Write the equation, find the cost of an 8-mile ride, and name the y-intercept's meaning.

  1. Starting value b = 3.50; rate m = 2.25 per mile → C = 2.25d + 3.50.
  2. d = 8: C = 2.25(8) + 3.50 = 18 + 3.50 = 21.50.
  3. The y-intercept (0,\ 3.50) is the cost before driving any miles — the flat fee.
Graph of taxi cost C = 2.25d + 3.50 against miles d, from 0 to 10 miles. The line starts at the y-intercept (0, 3.50), the flat fee before driving. Each mile adds 2.25 dollars (dashed step from 2 to 3 miles). At 8 miles the cost is 21.50 dollars, marked at (8, 21.50).123456789104812162024280Miles driven (d)Cost C ($)1 mi$2.25(0, 3.50) flat fee(8, 21.50)C = 2.25d + 3.50

Answer: C = 2.25d + 3.50; an 8-mile ride costs $21.50.

Check: 21.50 - 3.50 = 18 = 2.25 \times 8 ✓

Worked example

Example 4 — Equation from a graph (real world)

Problem. A gym's cost graph passes through (0, 50) and (4, 150) (months, dollars). Write the equation and find the cost for a year.

Graph of total gym cost in dollars (vertical axis, 0 to 250, grid every 25) versus months of membership (horizontal axis, 0 to 6). A straight line starts at the marked point (0, 50) and passes through the marked point (4, 150).0123456050100150200250Months of membershipTotal cost (dollars)(0, 50)(4, 150)
  1. b = 50 (sign-up fee).
  2. m = \frac{150 - 50}{4 - 0} = 25 dollars per month.
  3. C = 25m + 50. For 12 months: 25(12) + 50 = 350.
Graph of gym cost C = 25m + 50 against months, from 0 to 12 months. The line starts at (0, 50), the sign-up fee. A dashed triangle from (0, 50) to (4, 150) shows a run of 4 months and a rise of 100 dollars, so m = 100/4 = 25 dollars per month. At 12 months the cost is 350 dollars, marked at (12, 350).123456789101112131001502002503003504000Months (m)Cost C ($)run 4 monthsrise $100(0, 50)(4, 150)(12, 350)100/4 = $25 per month

Answer: C = 25m + 50; one year costs $350.

Check: At 4 months: 25(4) + 50 = 150 ✓

Worked example

Example 5 — GED-hard: from two points, work backwards

Problem. A line passes through (3, 1) and (-6, 7). Find its equation and its x-intercept.

  1. m = \frac{7 - 1}{-6 - 3} = \frac{6}{-9} = -\frac{2}{3}.
  2. Use (3, 1): 1 = -\frac{2}{3}(3) + b \Rightarrow 1 = -2 + b \Rightarrow b = 3. So y = -\frac{2}{3}x + 3.
  3. x-intercept: 0 = -\frac{2}{3}x + 3 \Rightarrow \frac{2}{3}x = 3 \Rightarrow x = 3 \times \frac{3}{2} = 4.5.
Coordinate plane from −7 to 6 by −2 to 8. A line passes through (−6, 7) and (3, 1). A dashed right triangle shows run 9 and rise −6, so m = −6/9 = −2/3. The line crosses the y-axis at (0, 3) and the x-axis at (4.5, 0): y = −(2/3)x + 3.−7−5−4−3−2−1126−2−1245680xyrun = 9rise = −6(−6, 7)(3, 1)(0, 3)(4.5, 0)y = −(2/3)x + 3

Answer: y = -\tfrac{2}{3}x + 3; x-intercept (4.5,\ 0).

Check: -\frac{2}{3}(-6) + 3 = 4 + 3 = 7 ✓ and -\frac{2}{3}(4.5) + 3 = -3 + 3 = 0 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

  • Use the calculator to fill a table of values quickly: e.g. for y = 2.25x + 3.5 type 2.25 × 8 + 3.5 enter → 21.5.
  • For fractional slopes, enter them with n/d: n/d 2 ▼ 3 ▶ × 9 − 1 enter → 5. Use toggle to switch fraction ↔ decimal.
  • Use the (−) key for negative coordinates: -2(-2) is (−) 2 × (−) 2.
  • When NOT to use it: plotting points, counting rise and run, or finding intercepts of equations like 3x + 4y = 24 — those are one-step mental divisions. The calculator can't draw the graph for you on the GED.
Common mistake

Common mistakes

  • Starting at the wrong place. Plotting b on the x-axis: (3, 0) instead of (0, 3). ✗ The y-intercept is always (0, b). ✓
  • Rise and run swapped. For m = -\frac{2}{5}, moving down 5 and right 2. ✗ Numerator = up/down, denominator = across. ✓
  • Losing a negative. In 5x - 2y = 10, setting x = 0 gives -2y = 10, so y = -5, not 5. ✓
  • Swapping m and b. Writing y = 2x - \frac{3}{4} for a line with slope -\frac{3}{4} and intercept 2. ✗ The coefficient of x is the slope. ✓
  • Treating every line as proportional. Using \frac{150}{4} as the slope when the line starts at $50. ✗ Slope uses two points: \frac{150 - 50}{4 - 0}. ✓
  • Plotting only two points. One small error makes a wrong line. ✗ Plot a third point as a check. ✓
Test strategy

GED test strategy

  • Match a graph to an equation by elimination. Check the y-intercept first (it rules out choices fast), then the sign of the slope, then the size.
  • Plug points in. When answer choices are equations, test 2 points from the graph in each — intercepts give the easiest arithmetic.
  • Use the formula sheet for y = mx + b and point-slope form when writing an equation.
  • Read real-world graphs carefully: check the axis labels and scale; identify the starting value (intercept) and rate (slope) with units.
  • Hot-spot graphing: work out the exact coordinates on your erasable note board before clicking.
  • Time: these questions should take 1–2 minutes. If an equation is not in y = mx + b form, rewrite it first — it saves time overall.

Check yourself

Graph linear equationsNo calculator

Where does the graph of y = 4x - 5 cross the y-axis?

Answer choices
Hint

On the y-axis, x = 0. What is y when x = 0?

Check yourself

Graph linear equationsHot spot

Click the point where the graph of y = -x + 3 crosses the x-axis.

Click on the figure to mark your answer (or focus it and use the arrow keys).

Coordinate plane from -5 to 5 on both axes-5-5-4-4-3-3-2-2-1-11122334455xy
Hint

On the x-axis, y = 0. Set y = 0 and solve for x.

Summary

  • y = mx + b: plot (0, b), then use rise/run from the slope to plot more points.
  • Ax + By = C: find the intercepts — y = 0 gives the x-intercept, x = 0 gives the y-intercept.
  • A point is on a line exactly when its coordinates make the equation true.
  • From a graph: read b, compute m from two grid points, write y = mx + b. From two points: find m, then solve for b.
  • In real life, b is the starting value and m is the rate per unit.
  • Horizontal lines are y = k; vertical lines are x = k.
Practice this skill →