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Writing Linear Equations from a Situation

Translate words into one- and two-variable linear equations, and rearrange a formula to solve for any letter.

Practice this skill 25 min Write linear equations from context Leer en español
Key idea

What you'll learn

Many GED questions don't ask you to solve anything. Instead they ask: "Which equation can be used to find…?" or "Which equation represents this situation?" In this lesson you will learn to:

  • translate a sentence like "seven less than three times a number is 20" into 3n - 7 = 20;
  • write a two-variable equation such as C = 150 + 22g or 1.25x + 0.80y = 50 from a real situation;
  • solve a formula for a different letter — for example, turn I = prt into p = \dfrac{I}{rt}, or P = 2l + 2w into w = \dfrac{P - 2l}{2}.

Why it matters on the GED: "Which equation models…" questions are common, and the answer choices are designed so that one small error (a flipped order, a missing parenthesis, a swapped rate) leads to a wrong choice. Rearranging formulas lets you use the GED formula sheet backwards — finding a height from a volume, or a rate from the interest earned.

Real life: writing an equation is how you build a budget rule ("$40 a day plus $0.25 a mile"), compare offers, or set up a spreadsheet formula.

Words to know

  • expression — numbers, letters, and operations with no equals sign (3n - 7).
  • equation — two expressions joined by = (3n - 7 = 20).
  • variable — a letter for a quantity that can change or is unknown (g = number of guests).
  • constant / starting value — an amount that does not change (a $150 setup fee).
  • rate of change — the amount added for each one unit ($22 per guest); it multiplies the variable.
  • two-variable equation — an equation with two unknowns, such as C = 150 + 22g; every pair (g, C) that makes it true is a solution.
  • literal equation (formula) — an equation with several letters, such as d = rt.
  • solve for a variable — rearrange so that one letter is alone on one side (t = \tfrac{d}{r}).
  • coefficient — the number (or letter) multiplying a variable (in \tfrac{1}{2}bh, the coefficient of h is \tfrac{1}{2}b).
  • reciprocal — a fraction flipped over (the reciprocal of \tfrac{1}{3} is 3; of \tfrac{5}{9} is \tfrac{9}{5}).

All Math words in the glossary

Translating sentences into one-variable equations

Read the sentence in chunks and translate each chunk. The words "is", "equals", "was", "will be" become =.

WordsExpression
a number increased by 6n + 6
6 more than a numbern + 6
6 less than a numbern - 6 (not 6 - n)
a number less than 66 - n
twice a number2n
a third of a number\tfrac{n}{3} or \tfrac{1}{3}n
twice the sum of a number and 62(n + 6)
twice a number, plus 62n + 6
the product of 4 and a number, decreased by 94n - 9

Watch "less than" and "more than". In "7 less than 3n", you start with 3n and take 7 away: 3n - 7. The order flips compared with the words.

Watch "the sum of" / "the difference of". Those words group things: "three times the sum of n and 4" is 3(n + 4), but "three times n, plus 4" is 3n + 4.

Example: "Five more than twice a number is 31." → 2n + 5 = 31, so 2n = 26 and n = 13. Check: 2(13) + 5 = 31 ✓

Hidden relationships. "Maria is 4 years older than twice her son's age s" → Maria's age = 2s + 4. "Pat has $12 less than Lee" → if Lee has L, Pat has L - 12.

Writing two-variable equations from a context

Pattern 1 — start value + rate. When something starts at a fixed amount and changes by the same amount for each unit, write:

y = (\text{rate})\,x + (\text{start})
  • A caterer charges $150 setup plus $22 per guest: C = 22g + 150.
  • A 15-inch candle burns 0.5 inch per hour: h = -0.5t + 15 (the rate is negative because the height goes down).
  • An account starts with $400 and you withdraw $20 per week: B = 400 - 20w.
Graph of a burning candle, h = −0.5t + 15. Horizontal axis: time t in hours from 0 to 30. Vertical axis: height h in inches from 0 to 16. The line starts at (0, 15), the starting height, and falls 0.5 inch per hour, reaching (30, 0) when the candle is gone. A slope triangle shows that 2 hours lower the height by 1 inch.04812162024280246810121416Time (t, hours)Height (h, inches)(0, 15) start(30, 0)+2 h−1 inh = −0.5t + 15

Ask yourself: Which number happens once? (the constant) and Which number happens again for every unit? (the coefficient of the variable). Swapping these is the most common wrong answer choice.

Pattern 2 — two kinds of items with a total. When you buy x of one thing and y of another:

(\text{price}_1)\,x + (\text{price}_2)\,y = \text{total}

Notebooks at $1.25 and pens at $0.80, spending $50: 1.25x + 0.80y = 50. Each price stays with its own item.

Pattern 3 — repeated pieces. If a rental is $40 per day for 3 days plus $0.25 per mile: 3(40) + 0.25m = 172.50. The daily charge is multiplied by the number of days; the per-mile charge is multiplied by the miles.

Test a choice with numbers. If you are unsure whether C = 22g + 150 or C = 150g + 22 is right, try g = 1 guest. The first gives $172 (setup + one guest) — sensible. The second gives $172 too! Try g = 2: the first gives $194, the second $322. Only the first adds $22 per extra guest, so it is correct.

Solving a formula for a variable

To solve a formula for one letter, treat every other letter as if it were a number and use the same steps as for any equation: undo addition/subtraction first, then multiplication/division.

Example A: Solve d = rt for t. The t is multiplied by r, so divide both sides by r: t = \dfrac{d}{r}.

Example B: Solve P = 2l + 2w for w.

Labelled rectangle for the perimeter formula. The two long sides are each l (length) and the two short sides are each w (width). Going all the way around gives P = l + w + l + w = 2l + 2w; solving for w gives w = (P − 2l)/2.l (length)lw (width)wP = 2l + 2w(all four sides added)P − 2l = 2w → w = (P − 2l) / 2
  1. Subtract 2l from both sides: P - 2l = 2w.
  2. Divide both sides by 2: w = \dfrac{P - 2l}{2} (which also equals \tfrac{P}{2} - l).

Example C: Solve V = \tfrac{1}{3}\pi r^2 h for h.

Labelled cone for the formula V = (1/3)πr²h. The radius r runs from the center of the circular base to its edge. The height h runs straight up from the center of the base to the tip, meeting the radius at a right angle. Solving for h gives h = 3V/(πr²).h(height)rr = radiusV = (1/3)πr²h → 3V = πr²h→ h = 3V / (πr²)
  1. Multiply both sides by 3: 3V = \pi r^2 h.
  2. Divide both sides by \pi r^2: h = \dfrac{3V}{\pi r^2}.

Example D: Solve 3x - 4y = 12 for y.

  1. Subtract 3x from both sides: -4y = 12 - 3x.
  2. Divide every term by -4: y = \dfrac{12}{-4} - \dfrac{3x}{-4} = -3 + \dfrac{3}{4}x, so y = \dfrac{3}{4}x - 3.

Two ways to use a formula with numbers. You can rearrange first and then substitute, or substitute first and then solve. Both give the same answer. With I = prt, I = 180, p = 1500, t = 3: substituting first gives 180 = 1500 \cdot r \cdot 3 = 4500r, so r = 0.04 (4%). Substituting first is often safer when the numbers are simple.

Formula

Formulas you will rearrange

These formulas are on the GED formula sheet, so you never need to memorize them — but you do need to rearrange them:

P = 2l + 2w \qquad A = \tfrac{1}{2}bh \qquad A = \tfrac{1}{2}h(b_1 + b_2) \qquad V = \tfrac{1}{3}\pi r^2 h \qquad I = prt \qquad d = rt
  • P = perimeter, l = length, w = width; A = area, b = base, h = height; b_1, b_2 = the two parallel bases of a trapezoid.
  • V = volume of a cone, r = radius; I = interest, p = principal, r = rate (decimal), t = time in years; d = distance, r = speed.
  • The slope-intercept form y = mx + b (on the sheet) is the model for "start value + rate": m = rate, b = starting value.

Not on the formula sheet: temperature conversions such as F = \tfrac{9}{5}C + 32 and C = \tfrac{5}{9}(F - 32). If the GED needs them, it gives them in the question.

Worked example

Example 1 (easy): translate and solve

"Seven less than three times a number is 20." Write an equation and find the number.

  1. "Three times a number": 3n.
  2. "Seven less than" that: 3n - 7.
  3. "is 20": 3n - 7 = 20.
  4. Solve: add 7 → 3n = 27; divide by 3 → n = 9.
Bar model for 'seven less than three times a number is 20'. Top bar: three equal blocks labeled n, total 3n. Bottom bar, the same length: a 20 block followed by a dashed 7 block that is taken away. So 3n − 7 = 20, 3n = 27, and n = 9.3nnnn3n − 720− 7taken awaywhat is left is 203n − 7 = 20 → 3n = 27 → n = 9 (each block is 9)

Answer: 3n - 7 = 20; the number is 9

Check: 3(9) - 7 = 27 - 7 = 20 ✓

Worked example

Example 2 (real world): start value + rate

A caterer charges a $150 setup fee plus $22 per guest. Write an equation for the total cost C for g guests, then find the cost for 40 guests.

  1. The setup fee happens once → constant 150.
  2. $22 happens for every guest → 22g.
  3. Equation: C = 22g + 150.
  4. For 40 guests: C = 22(40) + 150 = 880 + 150 = 1030.
Graph of the catering cost C = 22g + 150. Horizontal axis: guests g from 0 to 50. Vertical axis: cost in dollars from 0 to 1,400. The line starts at (0, 150), the 150 dollars setup fee, and rises 22 dollars per guest; a slope triangle shows 10 more guests add 220 dollars. The point (40, 1030) is marked: 40 guests cost 1,030 dollars.01020304050$0$200$400$600$800$1000$1200$1400Guests (g)Total cost C(0, 150) setup fee(40, 1030)+10 guests+$220C = 22g + 150

Answer: C = 22g + 150; 40 guests cost $1,030

Check: one more guest (41) should add $22: 22(41) + 150 = 1052 = 1030 + 22 ✓

Worked example

Example 3 (real world): two kinds of items

A bake sale sells muffins for $2.50 and cookies for $1.25. It took in $200. Write an equation using m muffins and c cookies. If 48 muffins were sold, how many cookies were sold?

  1. Money from muffins: 2.50m. Money from cookies: 1.25c.
  2. Equation: 2.50m + 1.25c = 200.
  3. Substitute m = 48: 2.50(48) + 1.25c = 200, so 120 + 1.25c = 200.
  4. Subtract 120: 1.25c = 80. Divide by 1.25: c = 64.
Bar model for the bake sale. The total bar, 200, dollars splits into 120 dollars from 48 muffins at 2.50 dollars each and 80 dollars from cookies at 1.25 dollars each. Since 1.25c = 80, c = 64 cookies.total $200 = 2.50m + 1.25cmuffins: 2.50 × 48= $120cookies: 1.25c= $8080 ÷ 1.25 = 642.50(48) + 1.25c = 200 → 1.25c = 80 → c = 64

Answer: 2.50m + 1.25c = 200; 64 cookies

Check: 2.50(48) + 1.25(64) = 120 + 80 = \$200 ✓

Worked example

Example 4 (work backwards): find the interest rate

Solve I = prt for r. Then find the yearly rate if $1,500 earned $180 of simple interest in 3 years.

  1. r is multiplied by p and t. Divide both sides by pt: r = \dfrac{I}{pt}.
  2. Substitute: r = \dfrac{180}{1500 \times 3} = \dfrac{180}{4500} = 0.04.
  3. Change the decimal to a percent: 0.04 = 4\%.
Diagram of simple interest. A principal of 1,500 dollars earns the same interest each year for 3 years: three blocks of 60, dollars one per year, total I = 180 dollars. Each year's 60 dollars is 1,500 × r, so r = 60 ÷ 1,500 = 0.04, or 4%.interest I = $180 over t = 3 yearsyear 1$60year 2$60year 3$601,500 × r = 60r = I ÷ (pt) = 180 ÷ (1,500 × 3) = 180 ÷ 4,500 = 0.04 = 4%p = $1,500 (principal) · t = 3 years · I = $180

Answer: r = \dfrac{I}{pt}; the rate is 4% per year

Check: I = 1500 \times 0.04 \times 3 = 60 \times 3 = \$180 ✓

Worked example

Example 5 (GED-hard): rearrange, then use

The formula C = \tfrac{5}{9}(F - 32) converts Fahrenheit to Celsius. Solve it for F, then find F when C = 35.

  1. Multiply both sides by the reciprocal of \tfrac{5}{9}, which is \tfrac{9}{5}: \tfrac{9}{5}C = F - 32.
  2. Add 32 to both sides: F = \tfrac{9}{5}C + 32.
  3. Substitute C = 35: F = \tfrac{9}{5}(35) + 32 = 63 + 32 = 95.
Two side-by-side temperature scales lined up so equal heights mean equal temperatures. Celsius on the left, Fahrenheit on the right. 0 °C lines up with 32 °F, 100 °C with 212 °F, and the highlighted line shows 35 °C = 95 °F, from F = (9/5)C + 32.Celsius (°C)Fahrenheit (°F)020406080100501001502000 °C32 °Fwater freezes100 °C212 °Fwater boils35 °C95 °F35 °C = 95 °FF = (9/5)(35) + 32 = 63 + 32 = 95

Answer: F = \tfrac{9}{5}C + 32; 35°C is 95°F

Check: \tfrac{5}{9}(95 - 32) = \tfrac{5}{9}(63) = 35 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

Writing an equation is a reading skill — the calculator can't help with the set-up. It helps with:

  • Testing a choice. Pick a simple value (like 2 guests) and evaluate each answer choice to see which matches the story.
  • Fractions in formulas. For \tfrac{9}{5}(35) + 32, press n/d, type 9, move down, type 5, move right out of the fraction, then × 35 + 32 enter → 95.
  • Order of operations with a rearranged formula. For r = \dfrac{180}{1500 \times 3}, use parentheses: 180 ÷ ( 1500 × 3 ) enter → 0.04. Without the parentheses, the calculator does 180 \div 1500 and then multiplies by 3, which is wrong.
  • π formulas: use the π key rather than typing 3.14, unless the question tells you to use 3.14.
Common mistake

Common mistakes

  • Reversing "less than". "7 less than 3n" — wrong: 7 - 3n. Right: 3n - 7.
  • Missing parentheses. "Twice the sum of n and 4" — wrong: 2n + 4. Right: 2(n + 4).
  • Swapping the rate and the constant. "$3.50 plus $2.25 per mile" — wrong: C = 3.50m + 2.25. Right: C = 2.25m + 3.50. The number that repeats per mile multiplies m.
  • Dividing only one term when solving a formula. From P - 2l = 2w, wrong: w = P - l. Right: divide the whole side by 2: w = \dfrac{P - 2l}{2}.
  • Undoing a fraction the wrong way. From V = \tfrac{1}{3}\pi r^2 h, wrong: h = \dfrac{V}{3\pi r^2}. Right: multiply by 3 first: h = \dfrac{3V}{\pi r^2}.
  • Sign slips when dividing by a negative. From -4y = 12 - 3x, every term changes sign: y = -3 + \tfrac{3}{4}x.
Test strategy

GED test strategy

  • Plug in a test number. Choose easy values for the variables, work out the answer from the story, and see which equation gives the same result. This beats every "trap" choice.
  • Check the constant and the rate separately. Cover up the variable term: does the constant match the one-time amount? Then cover up the constant: does the coefficient match the per-unit amount?
  • Look at signs. Decreasing quantities (draining, spending, burning) need a negative rate or subtraction.
  • For "solve the formula for…": substitute numbers into the original formula and into your rearranged version. If they agree, you are right. Example: in P = 2l + 2w use l = 5, w = 3, P = 16; then \tfrac{16 - 10}{2} = 3 ✓.
  • Use the formula sheet, but read the letters carefully: r is the radius in V = \pi r^2 h but the rate in d = rt and I = prt.

Check yourself

Write linear equations from contextNo calculator

Which equation represents the sentence below?

"Seven less than three times a number n is 20."

Answer choices
Hint

Find "three times a number" first. Then take 7 away from it.

Check yourself

Write linear equations from context

A taxi company charges $3.50 to start a ride plus $2.25 per mile. Which equation gives the total cost C, in dollars, of a ride of m miles?

Answer choices
Hint

Which amount is paid only once? Which amount is paid again for every mile?

Summary

  • Translate in chunks; "is" means =; "less than" and "more than" flip the order; "sum of" needs parentheses.
  • Start value + rate: y = (\text{rate})x + (\text{start}); a decreasing amount has a negative rate.
  • Two kinds of items: (\text{price}_1)x + (\text{price}_2)y = \text{total}.
  • To solve a formula for a letter, treat the other letters as numbers: undo +/- first, then \times/\div, and divide the whole side.
  • Test any equation with a simple number to make sure it matches the story.
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