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MTH1W Solving Linear Equations
Grade 9 Unit 3 — practice questions with full solutions

The balance method, multi-step equations, equations with fractions and decimals, rearranging formulas, and turning word problems into equations.

MTH1W · Grade 9 Strand C — Algebra 12 core 1 challenge
1Core
Solve:   5x − 7 = 18
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  1. Add 7 to both sides: 5x = 25.
  2. Divide both sides by 5: x = 5.
  3. Check: 5(5) − 7 = 25 − 7 = 18. ✔
Final answerx = 5
2Core
Solve:   3(x − 4) = 2x + 1
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  1. Expand the left side: 3x − 12 = 2x + 1.
  2. Subtract 2x from both sides: x − 12 = 1.
  3. Add 12: x = 13.
  4. Check: 3(13 − 4) = 27 and 2(13) + 1 = 27. ✔
Final answerx = 13
3Core
Solve:   x⁄4 + 3 = 7
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  1. Subtract 3 from both sides: x⁄4 = 4.
  2. Multiply both sides by 4: x = 16.
Final answerx = 16
4Core
xxx3 unknown boxes + 2 unit weights14 unit weightsThe scale is perfectly balanced
The balance scale above is level. Each blue box has the same unknown mass x, and each yellow circle is 1 unit. (a) Write an equation for the scale. (b) Solve for x. (c) Explain what ‘subtracting 2 from both sides’ means in terms of the picture.
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  1. (a) The left pan holds 3 boxes and 2 unit weights: 3x + 2. The right pan holds 14 unit weights. Since it balances: 3x + 2 = 14.
  2. (b) Subtract 2 from both sides: 3x = 12.
  3. Divide both sides by 3: x = 4.
  4. (c) It means removing 2 unit weights from each pan at the same time — the scale stays balanced only if you do the same thing to both sides.
  5. Check: 3(4) + 2 = 14. ✔
Final answer(a) 3x + 2 = 14   (b) x = 4   (c) Removing 2 weights from both pans keeps the scale level
5Core
Solve:   (2x − 1)⁄3 = (x + 4)⁄2
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  1. Cross-multiply (or multiply both sides by 6): 2(2x − 1) = 3(x + 4).
  2. Expand both sides: 4x − 2 = 3x + 12.
  3. Subtract 3x: x − 2 = 12.
  4. Add 2: x = 14.
Final answerx = 14
6Core
Solve:   4 − 2(3x + 1) = 5x − 20
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  1. Expand the bracket, watching the negative: 4 − 6x − 2 = 5x − 20.
  2. Simplify the left side: 2 − 6x = 5x − 20.
  3. Add 6x to both sides: 2 = 11x − 20.
  4. Add 20: 22 = 11x, so x = 2.
Final answerx = 2
7Core
Rearrange C = 2πr to solve for r, then rearrange A = 1⁄2bh to solve for h.
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  1. For the first: divide both sides by 2π to isolate r.
  2. r = C⁄2π.
  3. For the second: multiply both sides by 2 first, giving 2A = bh.
  4. Then divide by b.
Final answerr = C⁄2π    and    h = 2A⁄b
8Core
(2x + 10)°(x + 50)°(3x)°ABC
Using the labelled triangle above, (a) write an equation using the angle sum of a triangle, (b) solve for x, and (c) state the size of all three angles.
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  1. (a) The three interior angles of any triangle add to 180°: (2x + 10) + (3x) + (x + 50) = 180.
  2. (b) Collect like terms: 6x + 60 = 180.
  3. Subtract 60: 6x = 120, so x = 20.
  4. (c) Substitute back: angle A = 2(20) + 10 = 50°; angle C = 3(20) = 60°; angle B = 20 + 50 = 70°.
  5. Check: 50 + 60 + 70 = 180. ✔
Final answer(a) 6x + 60 = 180   (b) x = 20   (c) A = 50°, C = 60°, B = 70°
9Core
The sum of three consecutive integers is 84. Find the integers.
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  1. Let the smallest integer be n. The next two are n + 1 and n + 2.
  2. Write the equation: n + (n + 1) + (n + 2) = 84.
  3. Simplify: 3n + 3 = 84.
  4. Subtract 3 and divide by 3: n = 27.
  5. Check: 27 + 28 + 29 = 84. ✔
Final answer27, 28 and 29
10Core
A phone plan costs $25 per month plus $0.10 per GB of data. One month the bill was $37. How many GB were used?
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  1. Let g be the number of GB used.
  2. Model the situation: 25 + 0.10g = 37.
  3. Subtract 25: 0.10g = 12.
  4. Divide by 0.10: g = 120.
Final answer120 GB
11Core
A rectangle has a perimeter of 46 cm. Its length is 5 cm more than its width. Find its dimensions.
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  1. Let the width be w. Then the length is w + 5.
  2. Perimeter: 2(w + 5) + 2w = 46.
  3. Expand: 2w + 10 + 2w = 46, so 4w + 10 = 46.
  4. Solve: 4w = 36, so w = 9, and length = 14.
  5. Check: 2(14) + 2(9) = 28 + 18 = 46. ✔
Final answerwidth 9 cm, length 14 cm
12Core
Solve:   x⁄2 − x⁄5 = 3
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  1. Use a common denominator of 10 on the left: 5x⁄10 − 2x⁄10.
  2. That simplifies to 3x⁄10 = 3.
  3. Multiply both sides by 10: 3x = 30.
  4. Divide by 3: x = 10.
Final answerx = 10
13Challenge
Solve:   0.4(x − 3) + 0.2x = 1.8
Show the full solution
  1. Expand the bracket: 0.4x − 1.2 + 0.2x = 1.8.
  2. Collect the x terms: 0.6x − 1.2 = 1.8.
  3. Add 1.2 to both sides: 0.6x = 3.
  4. Divide by 0.6: x = 5.
  5. Check: 0.4(2) + 0.2(5) = 0.8 + 1 = 1.8. ✔
Final answerx = 5

MTH1W Solving Linear Equations — common questions

Short answers to the things students ask most about this unit.

How do you solve multi-step equations?

Expand any brackets, collect like terms on each side, move variables to one side and numbers to the other, then divide. Whatever you do to one side, do to the other.

How do you turn a word problem into an equation?

Name the unknown with a variable, write down what each quantity equals in terms of that variable, then find the sentence in the question that says two things are equal.

More MTH1W practice — grade 9 math

MTH1W Number SenseMTH1W Algebraic Expressions & Exponent LawsMTH1W Linear Relations & GraphingMTH1W Geometry & MeasurementAll MTH1W units
← Unit 2: Algebraic Expressions & Exponent LawsUnit 4: Linear Relations & Graphing →
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