Math Vale

⚔️ Algebra Kingdom · Algebra

Multi-Step Equations

Solve equations that need distributing, combining like terms, or moving variables to one side.

In short

  • Work in a fixed order: distribute, combine like terms, collect the variable on one side, then undo.
  • The number outside a bracket multiplies every term inside it, sign included.
  • A term reverses its sign when it crosses the equals sign, because moving it means subtracting it from both sides.
  • If the variable cancels, read what is left: a true statement means every number works, a false one means no number does.

One order of moves, every time

Multi-step equations look intimidating because there is so much on the page. The cure is a fixed routine, used in the same order no matter what the equation looks like:

1. Distribute — get rid of every bracket. 2. Combine — tidy each side separately by adding its like terms. 3. Collect — move all the variable terms to one side and all the numbers to the other. 4. Undo — you now have a two-step equation; finish it.

Nothing new is being invented here. Steps 1 and 2 only rewrite each side; steps 3 and 4 are the balance moves you already know.

Distributing over a bracket

3(x + 4) means three copies of the whole bracket: (x + 4) + (x + 4) + (x + 4) = 3x + 12.

The outside number multiplies every term inside, including the constant. An area model shows why: a rectangle of height 3 and width x + 4 splits into a 3-by-x piece and a 3-by-4 piece, total area 3x + 12.

Watch the signs. 3(x - 4) is 3x - 12, and -2(x + 5) is -2x - 10: the minus outside multiplies both terms inside as well.

Combining like terms

Terms are like when they carry exactly the same variable part. 7x and -3x are like terms; 7x and 7 are not, and neither are 7x and 7x2.

To combine, add the coefficients and keep the variable part: 7x - 3x = 4x. Plain numbers combine with plain numbers: 5 - 11 = -6.

The sign in front of a term belongs to that term and travels with it. In 10y - 58 - 9y - 111, the variable terms are 10y and -9y (giving y) and the constants are -58 and -111 (giving -169), so the side tidies to y - 169.

The variable on both sides

In 8x + 3 = 5x + 21 you cannot divide yet, because x appears twice. Gather the variable terms on one side first.

Subtract 5x from both sides — a legitimate balance move, since 5x is a genuine quantity: 3x + 3 = 21. Now it is an ordinary two-step equation, and x = 6.

A term that crosses the equals sign always reverses its sign, because "moving it" is really shorthand for subtracting it from both sides. Choosing to gather on the side with the larger coefficient keeps the numbers positive and the working tidy.

When there is no solution, or every solution

Not every equation has exactly one answer. Run the routine and see what is left after the variable terms cancel.

  • 3(x + 4) = 3x + 12 becomes 3x + 12 = 3x + 12, then 0 = 0. Every number works: infinitely many solutions.
  • 3(x + 4) = 3x + 7 becomes 3x + 12 = 3x + 7, then 12 = 7, which is false. No solution.
  • 3(x + 4) = 5x + 2 keeps a variable term, 12 - 2 = 5x - 3x, so x = 5. Exactly one solution.

The variable disappearing is not itself a verdict — look at the statement that is left and ask whether it is true or false.

Worked examples

Example 1

Solve for x: 4(x - 3) = 2x + 10

  1. Distribute the 4 over both terms in the bracket: 4x - 12 = 2x + 10.
  2. Collect the variable terms: subtract 2x from both sides, giving 2x - 12 = 10.
  3. Collect the numbers: add 12 to both sides, giving 2x = 22.
  4. Divide both sides by 2: x = 11.
  5. Check: 4(11 - 3) = 4 x 8 = 32, and 2 x 11 + 10 = 32. Both sides agree.

Example 2

Two stables hire out horses. The Silver Bridle asks 96 gold up front plus 4 gold a day; the Iron Gate asks 36 gold up front plus 9 gold a day. After how many days do they cost the same?

  1. Let n be the number of days, and write a cost expression for each stable.
  2. Silver Bridle: 96 + 4n. Iron Gate: 36 + 9n.
  3. Costing the same means setting the expressions equal: 96 + 4n = 36 + 9n.
  4. Subtract 4n from both sides and subtract 36 from both sides: 60 = 5n.
  5. n = 12 days. Check: 96 + 4 x 12 = 144 and 36 + 9 x 12 = 144. Correct.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Solve for x: 5x + 10 = x + 26

Answer: 4

  1. 5x + 10 = x + 26
  2. 4x + 10 = 26 (subtract x from both sides)
  3. 4x = 26 - 10 = 16 (subtract 10 from both sides)
  4. x = 16 / 4 = 4
  5. Check: left 5 x 4 + 10 = 30; right 1 x 4 + 26 = 30. Both sides agree.

Problem 2

Difficulty 3 of 5

Solve for w: 2(w + 12) = 42

Answer: 9

  1. 2(w + 12) = 42
  2. 2w + 24 = 42 (distribute the 2 over both terms)
  3. 2w = 42 - 24 = 18
  4. w = 18 / 2 = 9
  5. Check: 2(9 + 12) = 2 x 21 = 42. Correct.

Problem 3

Difficulty 4 of 5

Solve for w: 12w + 10 - 7w - 17 = -17

Answer: -2

  1. 12w + 10 - 7w - 17 = -17
  2. 5w - 7 = -17 (combine like terms)
  3. 5w = -17 - (-7) = -10 (add 7 to both sides)
  4. w = -10 / 5 = -2
  5. Check: 12 x (-2) + 10 - 7 x (-2) - 17 = -17. Correct.

Common mistakes

  • Distributing to the first term only: writing 3(x + 4) as 3x + 4.
  • Forgetting that a minus outside a bracket multiplies both terms, so -2(x + 5) becomes -2x + 10 instead of -2x - 10.
  • Moving a term across the equals sign without changing its sign.
  • Combining unlike terms, for example turning 5x + 3 into 8x.

What you should be able to do

  • Distribute and combine like terms before solving.
  • Solve equations with the variable on both sides.
  • Recognise equations with no solution or infinitely many solutions.
  • Translate multi-step word problems into equations.

Where this fits in the curriculum

Common Core

  • 8.EE.C.7.A

    Grade 8 — Give examples of linear equations with one solution, no solution or infinitely many solutions.

  • 8.EE.C.7.B

    Grade 8 — Solve linear equations with rational coefficients, expanding with the distributive property and collecting like terms.

  • HSA-REI.B.3

    High school — Solve linear equations and inequalities in one variable, including with letters for coefficients.

Ontario

  • G8.C2.3

    Grade 8 — Solve equations that involve multiple terms, integers and decimal numbers, and verify the solution.

  • MTH1W.C1.5

    Grade 9 de-streamed — Create and solve equations for various contexts, and verify their solutions.

SAT

  • Heart of Algebra

    Solving linear equations in one variable, including with literal coefficients.

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