Math Vale

⚔️ Algebra Kingdom · Algebra

Variables & Expressions

Use letters to stand for unknown or changing quantities, evaluate expressions, and combine like terms.

In short

  • A variable is a placeholder for a number; an expression has no equals sign, so it is evaluated or simplified rather than solved.
  • Substituted values always go in brackets — 5(9) is a product, while 59 is a different number entirely.
  • Only terms with identical letter parts can be combined, so 3x + 4 is already as simple as it gets.
  • A factor outside a bracket multiplies every term inside, and a negative factor flips every sign.
  • "5 less than n" is n - 5: some phrases reverse the order in which they are spoken.

A letter is a placeholder, not a mystery

A variable is a letter standing in for a number. Sometimes that number is unknown and waiting to be found; sometimes it is a number that changes, like the number of days a quest lasts.

An expression is a piece of mathematics with no equals sign: 3x + 7, or 2(n - 5). It does not claim anything, so it cannot be solved — it can only be evaluated for a value or simplified. An equation does claim something: 3x + 7 = 19 says two expressions are equal, and that can be solved.

The pieces of an expression have names worth knowing:

  • term — a chunk separated by + or -, such as 3x or 7
  • coefficient — the number multiplying the letter, the 3 in 3x
  • constant — a term with no letter at all

Writing 3x with nothing between the 3 and the x means multiply. The multiplication sign is dropped precisely because it looks too much like the letter x.

Evaluating: put the number back in

To evaluate an expression, replace each letter with its value and follow the order of operations.

Always put a substituted value in brackets. It costs one keystroke and prevents the two most common accidents.

  • 5x when x = 9 becomes 5(9) = 45. Writing 59 is not the same thing.
  • 3x2 when x = -4 becomes 3(-4)2. The exponent belongs to the x alone, so square first: (-4)2 = 16, then 3 x 16 = 48.

Notice that -x2 and (-x)2 are different. In -x2 the square happens before the minus sign is applied, giving -16 when x = 4; in (-x)2 the whole negative quantity is squared, giving 16.

Like terms: only identical letter parts combine

Two terms are like terms when their letter parts are exactly the same — the same letters raised to the same powers.

  • 3x and 5x are like terms: 3x + 5x = 8x
  • 3x and 4 are not: 3x + 4 stays as 3x + 4
  • 3x and 3y are not, and neither are x2 and x

Why can 3x + 4 not become 7x? Test it. If x = 2 then 3x + 4 = 10, but 7x = 14. The two are not the same expression, so the move was illegal.

Think of it as counting objects: 3 lanterns plus 5 lanterns is 8 lanterns, but 3 lanterns plus 4 coins cannot be merged into anything shorter. When collecting terms, the sign in front of a term always travels with it: in 7a - 2b - 3a, the term being collected with 7a is -3a, giving 4a - 2b.

The distributive property

The rule a(b + c) = ab + ac says that a factor outside a bracket multiplies every term inside it.

The area model shows why. A rectangle of height 4 whose base is split into a part of width x and a part of width 3 has total area 4(x + 3). Cut it down the split and you get two rectangles of area 4x and 12, so 4(x + 3) = 4x + 12.

Leaving the second term untouched — writing 4(x + 3) = 4x + 3 — is the single most common slip in algebra.

Watch the signs when the factor is negative. Multiplying by -2 flips the sign of everything inside:

  • -2(3x - 5) = -6x + 10

Expanding and then collecting like terms is how most multi-step algebra begins:

  • 3(x + 2) + 2(x - 5) = 3x + 6 + 2x - 10 = 5x - 4

Turning words into algebra

Modelling a situation means choosing a letter, saying clearly what it stands for, and translating the words in the order the operations actually happen.

The phrases that trip people up are the ones that reverse the spoken order:

  • "5 more than n" is n + 5 — order does not matter for addition
  • "5 less than n" is n - 5, not 5 - n
  • "n subtracted from 5" is 5 - n

Words like "the sum of" or "the difference of" gather a group together, and a group needs brackets before anything else is done to it: "3 times the sum of x and 4" is 3(x + 4), while "3 times x, plus 4" is 3x + 4.

A useful pattern for real situations is *fixed part + rate x quantity*. "A fee of 20 gold plus 5 gold per day" becomes 5d + 20, where d is the number of days. The 20 is paid once; the 5 is paid for each day.

Worked examples

Example 1

Evaluate 2a2 - 3b when a = -3 and b = 4.

  1. Substitute both values, using brackets: 2(-3)2 - 3(4).
  2. Exponents come first, and the square applies only to the -3: (-3)2 = 9.
  3. Now the multiplications: 2 x 9 = 18 and 3 x 4 = 12.
  4. Finally the subtraction: 18 - 12 = 6.
  5. Common trap avoided: (-3)2 is +9, not -9, because a negative times a negative is positive.

Example 2

Expand and simplify 5(2x + 3) - 2(x - 4).

  1. Expand the first bracket: 5 multiplies both terms, giving 10x + 15.
  2. Expand the second bracket: the factor is -2, so -2 x x = -2x and -2 x -4 = +8.
  3. The expression is now 10x + 15 - 2x + 8.
  4. Collect the x terms: 10x - 2x = 8x.
  5. Collect the constants: 15 + 8 = 23. The answer is 8x + 23.

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

Evaluate 2m + 6 when m = 5.

Answer: 16

  1. Substitute m = 5: 2(5) + 6
  2. 2(5) = 10
  3. 10 + 6 = 16

Problem 2

Difficulty 3 of 5

Simplify by combining like terms: 2s + 7 + 9s + 9

Answer: 11s + 16

  1. Mark the like terms: 2s and +9s carry s; 7 and 9 are plain numbers.
  2. s terms: 2 + 9 = 11
  3. Numbers: 7 + 9 = 16
  4. So the expression simplifies to 11s + 16.

Problem 3

Difficulty 4 of 5

Which expression means "9 less than 12 times a number r"?

  1. 9 - 12r
  2. 12r + 9
  3. 12(r - 9)
  4. 12r - 9

Answer: D. 12r - 9

  1. The phrase is "9 less than 12 times a number r".
  2. Name the unknown: let r be the number.
  3. Translate each piece in turn, keeping the order of operations the words describe.
  4. That gives 12r - 9.

Common mistakes

  • Combining unlike terms, writing 3x + 4 as 7x.
  • Reading 5x with x = 9 as the digits 59 instead of the product 45.
  • Distributing to only the first term inside a bracket: 4(x + 3) = 4x + 3.
  • Losing a sign when a negative factor multiplies a bracket, as in -2(3x - 5).
  • Squaring a negative incorrectly, treating (-4)2 as -16.
  • Translating "5 less than n" as 5 - n.

What you should be able to do

  • Translate a phrase in words into an algebraic expression.
  • Evaluate an expression for given values of its variables.
  • Combine like terms and apply the distributive property.
  • Tell the difference between an expression and an equation.

Where this fits in the curriculum

Common Core

  • 6.EE.A.2

    Grade 6 — Write, read and evaluate expressions in which letters stand for numbers.

  • 6.EE.A.3

    Grade 6 — Apply the properties of operations to generate equivalent expressions, including the distributive property.

  • 6.EE.A.4

    Grade 6 — Identify when two expressions are equivalent, whatever value the variable takes.

Ontario

  • G7.C2.1

    Grade 7 — Add and subtract monomials with a degree of 1 that involve whole numbers (collecting like terms).

  • G7.C2.2

    Grade 7 — Evaluate algebraic expressions that involve whole numbers and decimal numbers.

  • MTH1W.C1.4

    Grade 9 de-streamed — Simplify algebraic expressions by applying properties of operations of numbers, using various representations and tools.

SAT

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