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Writing the Equation of a Line from Its Slope and a Point

Use a slope and one point to write a line's equation in point-slope form and in slope-intercept form.

Practice this skill 25 min Line from slope and a point Leer en español
Key idea

What you'll learn

A straight line is completely decided by two facts: how steep it is (its slope) and one point it goes through. In this lesson you will turn those two facts into an equation.

You will learn to:

  • write a line in slope-intercept form, y = mx + b;
  • write a line in point-slope form, y - y_1 = m(x - x_1);
  • switch from point-slope form to slope-intercept form;
  • build an equation from a real-life story, such as "$60 per hour plus a fee."

Why it matters on the GED: The test often says something like "A line has a slope of -2 and passes through (-1, 4). Which equation represents the line?" It also hides the same question inside word problems about phone plans, taxi fares, savings, or a tank draining. Both forms of a line are printed on the GED formula sheet, so the real skill is knowing how to use them without sign mistakes.

Real life: Any time something changes at a steady rate (dollars per hour, miles per gallon, gallons per minute), a line equation lets you predict the future: "How much will a 7-hour job cost?" or "When will my gift card run out?"

Words to know

  • Ordered pair / point — a location written (x, y): first move left/right, then up/down. Example: (3, -2) is 3 right, 2 down.
  • Linear equation — an equation whose graph is a straight line. Example: y = 2x + 1.
  • Slope (m) — how steep a line is: the change in y for every 1 unit of change in x, or \dfrac{\text{rise}}{\text{run}}. Example: slope 3 means "up 3 for every 1 to the right."
  • Rate of change — slope in a word problem, such as $60 per hour or -15 gallons per minute.
  • y-intercept (b) — where the line crosses the y-axis; the point (0, b). Example: y = 2x + 5 crosses at (0, 5).
  • Starting value — the y-intercept in a word problem: the amount when x = 0, such as a flat fee.
  • x-intercept — where the line crosses the x-axis; found by setting y = 0.
  • Slope-intercept form — y = mx + b. Example: y = -4x + 7 has m = -4, b = 7.
  • Point-slope form — y - y_1 = m(x - x_1), where (x_1, y_1) is a known point. Example: y - 3 = 5(x - 2).
  • Distribute — multiply the number outside parentheses by every term inside. Example: -2(x + 1) = -2x - 2.

All Math words in the glossary

Slope-intercept form: y = mx + b

The most common way to write a line is

y = mx + b
  • m is the slope — the number multiplied by x.
  • b is the y-intercept — the number added on by itself.

Rule 1: If you know the slope and the y-intercept, just substitute.

Example: slope 4, y-intercept -3 → y = 4x - 3.

Example: slope -\tfrac{1}{2}, crosses the y-axis at (0, 6) → y = -\tfrac{1}{2}x + 6.

Rule 2: Watch the sign of b. A minus sign in front of the constant belongs to b.

Equationmb
y = 3x + 535
y = 3x - 53-5
y = -x + 2-12
y = \tfrac{2}{3}x\tfrac{2}{3}0
y = 707

Rule 3: Special lines.

  • Slope 0 means a horizontal line: y = b. Example: slope 0 through (4, 7) → y = 7.
  • A vertical line has an undefined slope and cannot be written as y = mx + b. Through (4, 7) it is x = 4.

A point (x, y) is on a line exactly when its numbers make the equation true. Is (2, 5) on y = 4x - 3? 4(2) - 3 = 5 ✓ Yes. This "plug in the point" test is your best checking tool.

Coordinate plane with the line y = 4x - 3. The y-intercept (0, -3) is marked. A green slope triangle goes right 1 and up 4 from (0, -3) to (1, 1), showing slope 4. The point (2, 5) is marked on the line, because 4(2) - 3 = 5.−3−2−12345−2−1123456780xyright 1up 4(0, −3)(2, 5)y = 4x − 3
Coordinate plane from -5 to 5 on both axes showing the line y = (2/3)x + 1. The points (-3, -1), (0, 1) and (3, 3) are marked on the line; it crosses the y-axis at (0, 1). A dashed slope triangle goes from (0, 1) right 3 units (run = 3) to (3, 1), then up 2 units (rise = 2) to (3, 3).-5-4-3-2-112345-5-4-3-2-1123450xyy = (2/3)x + 1run = 3rise = 2(-3, -1)(0, 1)(3, 3)

The line y = \tfrac{2}{3}x + 1. From the y-intercept (0, 1), go right 3 and up 2 to reach another point, (3, 3). The point (-3, -1) is on the same line: \tfrac{2}{3}(-3) + 1 = -2 + 1 = -1 ✓

Point-slope form: y − y₁ = m(x − x₁)

Often the GED gives you the slope and a point that is not the y-intercept. Point-slope form handles that directly:

y - y_1 = m(x - x_1)

Here (x_1, y_1) is the known point and m is the slope. The letters x and y stay as letters.

Rule 4: Put the point's numbers in with MINUS signs.

Example: slope 5 through (2, 3) → y - 3 = 5(x - 2).

Rule 5: A negative coordinate turns the minus into a plus. Subtracting a negative is adding.

SlopePointPoint-slope form
2(4, 1)y - 1 = 2(x - 4)
-3(-2, 5)y - 5 = -3(x + 2)
\tfrac{1}{2}(6, -4)y + 4 = \tfrac{1}{2}(x - 6)
-\tfrac{3}{4}(-8, -1)y + 1 = -\tfrac{3}{4}(x + 8)

Reading it backwards: In y + 4 = \tfrac{1}{2}(x - 6), the point is (6, -4) — flip both signs you see. The slope is the number in front of the parentheses, \tfrac{1}{2}.

Coordinate plane with the line y + 4 = (1/2)(x - 6). The known point (6, -4) is marked as (x1, y1). A green slope triangle goes right 2 and up 1 from (6, -4) to (8, -3), showing slope 1/2.−112345678910−8−6−5−4−3−2−1120xyright 2up 1(6, −4)y + 4 = ½(x − 6)

Different points on the same line give point-slope equations that look different but are the same line. Slope-intercept form looks the same whichever point you start from.

Getting to y = mx + b: two methods

Method A — Substitute and solve for b (usually fastest)

  1. Write y = mx + b with the known slope.
  2. Put the point's x and y in.
  3. Solve for b.
  4. Write the final equation with m and b.

Example: slope 3, point (2, 7).

7 = 3(2) + b \;\Rightarrow\; 7 = 6 + b \;\Rightarrow\; b = 1

So y = 3x + 1.

Method B — Start in point-slope form and simplify

  1. Write y - y_1 = m(x - x_1).
  2. Distribute m to both terms in the parentheses.
  3. Move y_1 to the right side (add or subtract it).

Example: slope -2, point (-1, 4).

y - 4 = -2(x + 1) y - 4 = -2x - 2 y = -2x + 2

Method A on the same problem: 4 = -2(-1) + b = 2 + b, so b = 2 ✓ Same answer.

Coordinate plane with the line y = -2x + 2, slope -2 through (-1, 4). A green slope triangle goes right 1 and down 2 from (-1, 4) to (0, 2), which is the y-intercept, so b = 2.−3−2−1234−4−3−2−115670xyright 1down 2(−1, 4)(0, 2), b = 2y = −2x + 2

Fractions: slope \tfrac{2}{3}, point (-6, 2). Method A: 2 = \tfrac{2}{3}(-6) + b = -4 + b, so b = 6 and y = \tfrac{2}{3}x + 6. Choosing an x that is a multiple of the denominator keeps the arithmetic whole.

Finding other facts once you have the equation:

  • x-intercept: set y = 0 and solve. For y = 3x - 6: 0 = 3x - 6, so x = 2; the point is (2, 0).
  • Missing coordinate: if (k, 10) is on y = 3x + 1, then 10 = 3k + 1, so k = 3.

Word problems: rate = slope, one known situation = point

In a story, look for two things:

  1. The rate (the words per, each, every) → that is m. If the amount goes down (draining, spending, losing), m is negative.
  2. One complete situation — an input together with its output, like "a 3-hour job costs $230" → that is a point (x, y) = (3, 230).

Then find b exactly as before. In context, b is the starting value: a flat fee, a sign-up charge, the amount in a tank at time 0.

Example: A plumber charges $60 per hour plus a flat fee. A 3-hour job costs $230.

  • m = 60, point (3, 230).
  • 230 = 60(3) + b = 180 + b, so b = 50.
  • Cost: C = 60h + 50. The flat fee is $50.

Big trap: the total ($230) is not the fee. The fee is what is left after you take away the hourly part.

Decreasing example: A tank drains at 15 gallons per minute and holds 380 gallons after 8 minutes. Then m = -15 and 380 = -15(8) + b = -120 + b, so b = 500: the tank started with 500 gallons, and W = -15t + 500.

Always name your variables with units ("h = hours, C = dollars") so you put the right numbers in the right places.

Formula

The formulas (both are on the GED formula sheet)

y = mx + b \qquad\qquad y - y_1 = m(x - x_1)
  • x, y — the variables (they stay letters in the final equation)
  • m — the slope (rate of change)
  • b — the y-intercept (value of y when x = 0)
  • (x_1, y_1) — one specific point on the line

Both slope-intercept form and point-slope form are printed on the GED formula sheet, along with the slope formula m = \dfrac{y_2 - y_1}{x_2 - x_1}. You do not need to memorize them — but you do need to memorize how to use them: plug the point in, solve for b. A handy shortcut that comes from Method A is b = y_1 - m x_1.

Worked example

Example 1 (easy): slope 5 through (2, 3)

Problem: Write the equation of the line with slope 5 through (2, 3) in point-slope form and in slope-intercept form.

  1. Point-slope form: put m = 5, x_1 = 2, y_1 = 3 into y - y_1 = m(x - x_1): y - 3 = 5(x - 2)
  2. Distribute: y - 3 = 5x - 10.
  3. Add 3 to both sides: y = 5x - 7.
Coordinate plane with the line y = 5x - 7, slope 5 through (2, 3). A green slope triangle goes right 1 and up 5 from (2, 3) to (3, 8). The line crosses the y-axis at (0, -7).−2−1123456−6−5−4−3−2−1124567890xyright 1up 5(2, 3)(0, −7)y = 5x − 7

Answer: y - 3 = 5(x - 2), or y = 5x - 7.

Check: Put x = 2 into y = 5x - 7: 5(2) - 7 = 3 ✓

Worked example

Example 2 (medium): a fraction slope and a negative point

Problem: A line has slope -\tfrac{2}{3} and passes through (-6, 1). Write its equation in slope-intercept form.

  1. Start with y = mx + b: y = -\tfrac{2}{3}x + b.
  2. Substitute x = -6, y = 1: 1 = -\tfrac{2}{3}(-6) + b
  3. Multiply: -\tfrac{2}{3} \times (-6) = +4 (negative times negative is positive). So 1 = 4 + b.
  4. Subtract 4: b = 1 - 4 = -3.
Coordinate plane with the line y = -(2/3)x - 3, slope -2/3 through (-6, 1). A green slope triangle goes right 3 and down 2 from (-6, 1) to (-3, -1), and the line crosses the y-axis at (0, -3).−8−7−6−5123−6−5−4−2−11230xyright 3down 2(−6, 1)(0, −3)y = −⅔x − 3

Answer: y = -\tfrac{2}{3}x - 3

Check: -\tfrac{2}{3}(-6) - 3 = 4 - 3 = 1 ✓

Worked example

Example 3 (word problem): an electrician's bill

Problem: An electrician charges $75 per hour plus a fixed trip fee. A 4-hour job costs $345. Write an equation for the cost C of an h-hour job, and find the cost of a 6-hour job.

  1. Rate → slope: m = 75.
  2. Known situation → point: (h, C) = (4, 345).
  3. Substitute into C = 75h + b: 345 = 75(4) + b = 300 + b.
  4. Solve: b = 345 - 300 = 45. The trip fee is $45.
  5. Equation: C = 75h + 45.
  6. 6-hour job: C = 75(6) + 45 = 450 + 45 = 495.
Graph of the electrician cost C = 75h + 45, hours on the horizontal axis from 0 to 7 and cost in dollars from 0 to 600. The line starts at the trip fee (0, 45) and passes through (4, 345) and (6, 495). A green slope triangle from (4, 345) to (6, 495) shows 2 more hours add 150 dollars, which is 75 dollars per hour.012345670100200300400500600Hours, hCost, C (dollars)2 hours+$150trip fee $45(4, 345)(6, 495)C = 75h + 45

Answer: C = 75h + 45; a 6-hour job costs $495.

Check: 4 hours: 75(4) + 45 = 345 ✓ matches the story.

Worked example

Example 4 (word problem, decreasing): a gift card

Problem: Maria spends $15 a week from a gift card. After 4 weeks she has $90 left. Write an equation for the balance B after w weeks. After how many weeks will the card be empty?

  1. Money goes down each week, so m = -15.
  2. Point: (w, B) = (4, 90).
  3. Substitute: 90 = -15(4) + b = -60 + b.
  4. Solve: b = 90 + 60 = 150. The card started at $150.
  5. Equation: B = -15w + 150.
  6. Empty means B = 0: 0 = -15w + 150 \Rightarrow 15w = 150 \Rightarrow w = 10.
Graph of the gift card balance B = -15w + 150, weeks from 0 to 11 across and dollars from 0 to 160 up. The line starts at (0, 150), passes through (4, 90), and reaches 0 at (10, 0) when the card is empty. A green slope triangle from (0, 150) to (4, 90) shows 4 weeks take away 60 dollars, which is -15 dollars per week.012345678911020406080100120140160Weeks, wBalance, B (dollars)4 weeks−$60(4, 90)empty:(10, 0)B = −15w + 150

Answer: B = -15w + 150; the card is empty after 10 weeks.

Check: 150 - 15 \times 10 = 0 ✓, and 150 - 15 \times 4 = 90 ✓

Worked example

Example 5 (GED-hard, work backwards): find the missing coordinate

Problem: A line with slope \tfrac{3}{4} passes through (8, -1). It also passes through (k, 5). Find k.

  1. Find b: -1 = \tfrac{3}{4}(8) + b = 6 + b, so b = -7.
  2. Equation: y = \tfrac{3}{4}x - 7.
  3. Put in the second point, y = 5, x = k: 5 = \tfrac{3}{4}k - 7.
  4. Add 7: 12 = \tfrac{3}{4}k.
  5. Multiply both sides by \tfrac{4}{3}: k = 12 \times \tfrac{4}{3} = 16.
Coordinate plane with the line y = (3/4)x - 7 through (8, -1) and (16, 5). A green slope triangle goes right 8 and up 6 from (8, -1) to (16, 5): 6/8 = 3/4. The line crosses the y-axis at (0, -7), and the second point gives k = 16.24618−6−4−22460xyright 8up 6(8, −1)(k, 5) = (16, 5)(0, −7)y = ¾x − 7

Answer: k = 16

Check: Slope from (8, -1) to (16, 5): \dfrac{5 - (-1)}{16 - 8} = \dfrac{6}{8} = \dfrac{3}{4} ✓

Calculator tip

Calculator tip: TI-30XS MultiView

Most of this skill is quick mental arithmetic, but the calculator helps when the numbers are decimals, fractions, or negatives.

  • Find b in one line with the shortcut b = y_1 - m \cdot x_1. For slope -2 through (-1, 4), type 4 − ( (−) 2 ) ( (−) 1 ) enter → 2. Use the gray (−) key for a negative number and the − key for subtraction — mixing them up can give a syntax error or a wrong answer.
  • Always put a negative number in parentheses when you multiply by it.
  • Fractions: use the n/d key to enter a slope like \tfrac{2}{3}. If the answer appears as a decimal and you want a fraction (or the other way round), press toggle.
  • Word problems with money: $0.35 per mile, 120 miles, total $56.50 → b = 56.50 - 0.35 \times 120 = 14.50.

When NOT to use it: writing point-slope form needs no arithmetic at all — just careful signs. And for whole-number problems like 7 = 3(2) + b, mental math is faster than typing.

Common mistake

Common mistakes

  • Using the point's y-value as b. Slope 3 through (2, 7) is not y = 3x + 7. The y-intercept is only the point where x = 0. Right: 7 = 6 + b, so y = 3x + 1.
  • Sign errors with negative coordinates. Through (-2, 5): wrong y - 5 = m(x - 2); right y - 5 = m(x + 2), because x - (-2) = x + 2.
  • Distributing to only the first term. Wrong: -2(x + 1) = -2x + 1. Right: -2(x + 1) = -2x - 2.
  • Negative times negative. In 1 = -\tfrac{2}{3}(-6) + b, the product is +4, not -4. Getting -4 makes b = 5 instead of -3.
  • Swapping x and y. The point (3, 5) means x_1 = 3, y_1 = 5. Wrong: y - 3 = 2(x - 5). Right: y - 5 = 2(x - 3).
  • Word problems: treating the total as the fee. "3 hours cost $230 at $60/hour" → the fee is 230 - 180 = 50, not $230. Also, a decreasing amount needs a negative slope.
Test strategy

GED test strategy

  • Eliminate by slope first. The slope is given to you — cross out any answer with a different number in front of x (watch the sign!). Usually two choices disappear immediately.
  • Then plug in the point. Put the given point into each remaining choice. Only the right line makes a true statement. This works for both forms and takes seconds.
  • Match the form they ask for. If every choice is in point-slope form, don't waste time converting — just check the slope and the signs of the point.
  • Read the word problem twice. Circle the rate ("per hour") and the full situation ("4 hours cost $345"). Ask: is the amount going up or down?
  • Sanity-check the meaning. A flat fee should be smaller than the total; a starting amount in a draining tank should be bigger than later amounts.
  • Budget your time: about 1–2 minutes per item.

Check yourself

Line from slope and a point

A line has a slope of 4 and passes through the point (0, -3).

Which equation represents the line?

Answer choices
Hint

The point (0, -3) is where the line crosses the y-axis. Which letter in y = mx + b stands for that?

Check yourself

Line from slope and a point

A line has a slope of 2 and passes through the point (3, 5).

Which equation represents the line in point-slope form?

Answer choices
Hint

Point-slope form is y - y_1 = m(x - x_1). Which number in (3, 5) is x_1, and which is y_1?

Summary

  • Slope-intercept form y = mx + b: m = slope, b = y-intercept (value of y when x = 0).
  • Point-slope form y - y_1 = m(x - x_1): put the point in with minus signs; a negative coordinate becomes a plus.
  • To get y = mx + b from a slope and a point: plug the point into y = mx + b and solve for b (or use b = y_1 - m x_1).
  • In word problems, the rate is the slope and one complete situation is the point; b is the starting value or fee. Going down means a negative slope.
  • Check every answer by plugging the point back in. Both forms are on the GED formula sheet.
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