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1.2.1

Linear Equations & Word Problems

EST I HS Math · Unit 1 · Foundational Algebra

Lesson 1.2 · Linear Equations & Word Problems

On the paper: Foundational Algebra is 27–32% of the paper; linear models appear 3–5 times on every real EST I

Big idea. A linear equation is a sentence with one unknown. The EST rarely wants the unknown itself — it wants the sentence read correctly, and something built from the unknown. Solve only as far as the question needs.

1Undo, in reverse order ↩️

An equation is the unknown wrapped in operations. To unwrap it, undo the operations in the opposite order they were applied — the last thing done is the first thing undone.

🍼 Baby steps

  1. 1

    3x − 5 = 10

    x was multiplied by 3, then 5 was subtracted. Undo the subtraction first.

  2. 2

    3x = 15

    Add 5 to both sides.

  3. 3

    x = 5

    Divide by 3. But look what the January 2023 paper asked next…

  4. 4

    10x + 5 = 55

    …it asked for 10x + 5, not x. Solving for x was only step one. Read the question to its last word.

Trap · the answer that is not the answer

In every real paper there is an option equal to x itself. Here that would be 5. It is there for the student who stops early.

2Words → equation, one phrase at a time 📝

🍼 Baby steps

  1. 1

    "a number" → x

    Name the unknown before reading further.

  2. 2

    "twice a number" → 2x "three times the number" → 3x

    "times" and "of" mean multiply.

  3. 3

    "the difference between twice a number and two" → 2x − 2

    "difference between A and B" is A − B, in the order written.

  4. 4

    "…is three times the number" → 2x − 2 = 3x

    "is" is the equals sign. That is the whole December 2021 question — they wanted the equation, not its solution.

  5. 5

    "seven years younger than x" → x − 7; "in five years" → x − 7 + 5 = x − 2

    Age questions: younger/older shift the number, "in n years" adds n to everyone (March 2021).

  6. 6

    consecutive even integers → n, n + 2, n + 4

    Consecutive = step 1; consecutive even or odd = step 2. (May 2021.)

3The cost model: fixed + rate × n 💵

Half of the EST's word problems are one line: C = (per-unit) × n + (fixed). The paper either asks you to build it, or shows it and asks what one number means.

🍼 Baby steps

  1. 1

    f = 7g + 88

    Oct 2020: a gaming site, g = games downloaded. Which number is "per game"? The one multiplied by g: 7.

  2. 2

    g = 0 → f = 88

    Set n to zero: what is still paid? 88 — the subscription, paid before a single download. That is the y-intercept.

  3. 3

    g = 1 → 95, g = 2 → 102

    Each extra game adds 7. The coefficient of n is the per-unit price — the slope.

  4. 4

    shirts: $5 per TWO + $2 wage each → 2.5r + 2r = 4.5r

    2026 sample: "for two shirts" hides a division. Per-shirt cost first, then multiply by r. No fixed part, so no "+ constant".

🎛 Play with it · fixed, rate, and break-even

Slide the three numbers. The shaded band is the fixed part (paid even at n = 0); the little step is the slope. Tick the second plan to see where two offers cost the same.

C = 5·n + 20n = 6 → C = 30 + 20 = 50

n (units)120fixed 20 — paid even when n = 0 (the y-intercept)+1 unit → +5 (the slope)

Rule · interpreting a number in a model

Multiplied by the variable → rate (per day / per game / per shirt). Standing alone → starting value (fee, deposit, price on day 0). The paper's wrong options swap these two.

4Change without solving (Δ) 📉

🍼 Baby steps

  1. 1

    10P + 2U = 2500

    Aug 2021: price P, units sold U. Price goes from 50 to 45 — how many MORE units?

  2. 2

    ΔP = −5

    You do not need U at 50 and U at 45. You need the change.

  3. 3

    10·ΔP + 2·ΔU = 0 → −50 + 2ΔU = 0

    The right-hand side is constant, so the changes balance.

  4. 4

    ΔU = 25

    25 more items. One line, no substitution.

5Real EST questions, solved twice 📄

Real question 1EST I HS · 2026 official sample · Q41Knowledge

(13/8)·a + 7x = 1/2

If the equation above holds and x = 1, what is the value of a?

  1. A.−2

    From (13/8)a = −6.5 → a = −6.5 ÷ 13 × 8 done as ÷ 8 × 13 gives −10.6; −2 is a guess.

  2. B.−3

    Sign slip: 7 − ½ instead of ½ − 7.

  3. C.−4
  4. D.−8

    Multiplied both sides by 8 but forgot the 13.

Safe way · ~40 s

  1. Put x = 1: (13/8)a + 7 = ½.
  2. (13/8)a = ½ − 7 = −13/2.
  3. a = (−13/2) × (8/13) = −4.

⏱ 90-second way · ~12 s

  1. ½ − 7 = −6.5 = −13/2. A 13 on both sides — cancel it: a/8 = −½ → a = −4.

Answer: C

Real question 2EST I · December 2021 · Section I · Q1Knowledge

The difference between twice a number and two is three times the number. Which of the following represents the equation that can be used to solve for the number?

  1. A.2x − 2 = 3(x − 2)

    "three times the number", not three times (the number minus two).

  2. B.2 − 2x = 3

    Order reversed and "three times the number" lost its x.

  3. C.2x − 2 = 3x
  4. D.2x − 3x = 3 + x

    Nothing in the sentence says "3 + x".

Safe way · ~30 s

  1. twice a number: 2x. "difference between 2x and two": 2x − 2.
  2. "is": =. "three times the number": 3x.
  3. 2x − 2 = 3x.

⏱ 90-second way · ~10 s

  1. Translate left to right, word by word. Do not solve — the question asks for the equation.

Answer: C

Real question 3EST I HS · 2026 official sample · Q9Reasoning

A company manufactures cotton T-shirts. The manufacturing cost for two shirts is $5 and the average worker's wage to produce one shirt is $2. Which equation best represents the total cost C of producing r shirts?

  1. A.C = 5r

    Uses $5 per shirt — but $5 is for TWO shirts — and forgets the wage.

  2. B.C = 5r − 2

    A wage is a cost; it is added, not subtracted, and it is per shirt, not once.

  3. C.C = 4.5r
  4. D.C = 2.5r + 2

    Right manufacturing rate, but the $2 wage is per shirt (× r), not a one-off fee.

Safe way · ~35 s

  1. Manufacturing per shirt: 5 ÷ 2 = 2.5.
  2. Wage per shirt: 2. Total per shirt: 4.5.
  3. C = 4.5r. No fixed cost is mentioned, so no constant.

⏱ 90-second way · ~12 s

  1. Everything is "per shirt" → the model is pure rate × r. Only one option has no stray constant and the right per-shirt number.

Answer: C

Real question 4EST I · October 2020 · Section II · Q28Application

A gaming website charges its clients by the number of games downloaded g. If the total fees f of a gamer are given by f = 7g + 88, what is the best interpretation of the term 88?

  1. A.The price of one downloaded game.

    That is the 7 — the number multiplied by g.

  2. B.The subscription fee, paid regardless of the number of games downloaded.
  3. C.The average number of games downloaded.

    g is the number of games; 88 is in dollars.

  4. D.The fees for downloading 7 games.

    7 games would cost 7 × 7 + 88 = 137.

Safe way · ~30 s

  1. Set g = 0: f = 88. Money paid with zero downloads.
  2. That is a fixed fee — the subscription.

⏱ 90-second way · ~8 s

  1. Standing alone = paid at n = 0 = fixed fee. Multiplied by g = per game.

Answer: B

Real question 5EST I · August 2021 · Section I · Q6Reasoning

An enterprise found that when the unit selling price P is raised, the number of units sold U goes down, modelled by 10P + 2U = 2500. The manager decided to decrease the unit selling price from $50 to $45. How many more items did he sell?

  1. A.75

    Would need ΔP = −15.

  2. B.50

    Forgot to divide by the 2 in front of U.

  3. C.25
  4. D.10

    The coefficient of P, copied.

Safe way · ~50 s

  1. P = 50: 500 + 2U = 2500 → U = 1000.
  2. P = 45: 450 + 2U = 2500 → U = 1025.
  3. 1025 − 1000 = 25.

⏱ 90-second way · ~12 s

  1. Only the change matters: 10·ΔP + 2·ΔU = 0 with ΔP = −5 → ΔU = 50 ÷ 2 = 25.

Answer: C

⏱ The 90-second lane

One question = 90 seconds. These are the moves that finish this topic in one line.

  • 1Read the last line first: is the question asking for x, or for 10x + 5, or for 5x − y? Solve only that far.
  • 2Words → symbols in the order written: "difference between A and B" = A − B, "is" = "=", "of"/"times" = ×.
  • 3"n years younger" → x − n; "in k years" → add k to every age.
  • 4Consecutive even/odd → n, n + 2, n + 4. Their sum is 3n + 6 = 3 × middle.
  • 5Cost models: the number multiplied by n is the rate (slope); the lone number is the fixed part (y-intercept, the value at n = 0).
  • 6"For two shirts" hides a ÷ 2. Convert to per-unit before multiplying by n.
  • 7Two plans equal? fixed₂ − fixed₁ = (rate₁ − rate₂) × n. Solve that one line.
  • 8Change questions: use Δ. The constant on the right disappears.

Take with you

  • ▸Undo operations in reverse order; then keep reading — the EST usually asks for an expression in x, not x.
  • ▸Translate a sentence phrase by phrase, left to right.
  • ▸C = rate·n + fixed: rate multiplies n, fixed survives at n = 0. Interpretation questions test exactly this swap.
  • ▸Δ trick: when only a change is asked, only the changes need to balance.

Keep reading — this lesson marks itself done automatically.