Jugami Pre-Algebra

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Expressions

The building blocks of algebra: numbers, letters and operations put together.

An expression is a math phrase like 3x + 12 or 2(a − 5). It has no equals sign. It is a recipe for a number: give the letters values and you can work it out.

This page names the parts of an expression, sets out the order of operations (and its left-to-right traps), and shows how to evaluate, write, simplify, distribute and factor expressions, and why those moves work. Then try the practice problems and the Expression Lab game.

Jump to practicePlay Expression Lab
An area model: a rectangle 3 tall and x + 4 wide, split into a 3 by x part with area 3x and a 3 by 4 part with area 12, showing 3(x + 4) = 3x + 12 3(x + 4) = 3x + 123xx1243area = 3x + 12

Terms, coefficients, constants and variables

A variable is a letter that stands for a number. Terms are the pieces joined by + or −. A coefficient is the number multiplying a variable (in 4x it is 4; in −2y it is −2; in x it is 1). A constant is a term with no variable, like 7.

The expression 4x + 7 − 2y split into three terms: 4x has coefficient 4 and variable x; 7 is the constant with no variable; −2y has coefficient −2 and variable y 4x + 7 − 2y: three terms4xcoefficient 4variable x+ 7constant 7(no variable)− 2ycoefficient −2variable y
4x + 7 − 2y has three terms. Read the sign in front of a term as part of it: the last term is −2y.

Example 1: Naming the parts

In 5a − 3 + b, list the terms, the coefficients and the constant.

  1. Terms: 5a, −3 and b (three terms).
  2. Coefficients: 5 (of a) and 1 (of b, since b = 1b).
  3. Constant: −3.

Order of operations

So everyone gets the same answer, mathematicians agree on an order: 1. Parentheses (and other grouping), 2. Exponents, 3. Multiplication and division, 4. Addition and subtraction. Some people remember this as PEMDAS.

The trap: PEMDAS lists M before D and A before S, but each pair has equal rank. Within a pair, work left to right. So 24 ÷ 4 × 2 = 6 × 2 = 12, and 10 − 3 + 2 = 7 + 2 = 9 (not 10 − 5).

Working 24 ÷ 4 × 2 from left to right: first 24 ÷ 4 = 6, then 6 × 2 = 12. Multiplying 4 × 2 first would wrongly give 3 × and ÷: left to right24 ÷ 4 × 2= 6 × 2= 12Not 24 ÷ 8 = 3: work left to right
Division and multiplication are a tie, so the one on the left goes first.

Example 2: Order of operations

Evaluate 20 − 2 × (3 + 4) and 18 ÷ 3 × 2.

  1. Parentheses first: 3 + 4 = 7, so 20 − 2 × 7.
  2. Multiply before subtracting: 2 × 7 = 14, and 20 − 14 = 6.
  3. 18 ÷ 3 × 2: left to right, 18 ÷ 3 = 6, then 6 × 2 = 12 (not 18 ÷ 6 = 3).

Exponents basics

An exponent says how many times to multiply a base by itself: 2⁴ = 2 × 2 × 2 × 2 = 16. Here 2 is the base and 4 is the exponent. Watch the parentheses: (−3)² = (−3) × (−3) = 9, but −3² means −(3 × 3) = −9.

Example 3: Exponents in an expression

Evaluate 5 + 2³ and (−4)² − 10.

  1. Exponents before adding: 2³ = 2 × 2 × 2 = 8, so 5 + 8 = 13.
  2. (−4)² = (−4) × (−4) = 16 (negative × negative is positive), so 16 − 10 = 6.

Evaluating expressions by substitution

To evaluate an expression, substitute (replace) each variable with its value, in parentheses, then follow the order of operations. Parentheses help with negatives; review the sign rules in Integers.

Example 4: Substituting a negative

Evaluate 3x² − 2x + 1 when x = −2.

  1. Substitute with parentheses: 3(−2)² − 2(−2) + 1.
  2. Exponent first: (−2)² = 4, so 3 × 4 − 2 × (−2) + 1.
  3. Multiply: 12 − (−4) + 1 = 12 + 4 + 1 = 17.

Writing expressions from words

Key words help: “sum” and “more than” mean +; “difference” and “less than” mean −; “product” and “times” mean ×; “quotient” and “per” mean ÷. Careful: “5 less than n” is n − 5, not 5 − n.

Example 5: Words to expressions

Write expressions for: (a) 5 less than twice a number n; (b) the cost of t tickets at $8 each plus a $4 fee. Find (b) when t = 6.

  1. (a) Twice n is 2n; 5 less than that is 2n − 5.
  2. (b) 8 dollars times t, plus 4: 8t + 4.
  3. When t = 6: 8 × 6 + 4 = 48 + 4 = $52.

Like terms and combining them

Like terms have the same variable part: 2x and −5x are like terms; 3 and 7 are like terms; 2x and 2x² are not, and 3x and 3y are not. To combine like terms, add their coefficients: 2x + x = 3x, just as 2 apples + 1 apple = 3 apples.

Algebra tiles: 2 x-tiles and 3 unit tiles plus 1 x-tile and 1 unit tile regroup into 3 x-tiles and 4 unit tiles, so 2x + 3 + x + 1 = 3x + 4 2x + 3 + x + 1 = 3x + 4xxx2x + 3+ x + 1xxx3x + 4group the x-tiles, group the unit tiles
Algebra tiles: a long tile is x and a small square is 1. Only tiles of the same kind combine.

Example 6: Combining like terms

Simplify 4x + 7 − 6x + 2 + y.

  1. Group like terms: (4x − 6x) + (7 + 2) + y.
  2. Add coefficients: 4 − 6 = −2, so −2x. Numbers: 9.
  3. Answer: −2x + y + 9. The y has no like term, so it stays.

The distributive property and factoring

The distributive property says a(b + c) = ab + ac: multiply the outside number by each term inside. An area model shows it: a rectangle a tall and b + c wide splits into two smaller rectangles.

Factoring goes the other way. Find the greatest common factor (GCF) of the terms, the same GCF you used to simplify in Fractions, and write it outside the parentheses.

An area model for 4(2x + 3): a rectangle 4 tall split into widths 2x and 3, with areas 8x and 12, so 4(2x + 3) = 8x + 12 4(2x + 3) = 8x + 128x2x12344 × 2x = 8x and 4 × 3 = 12
The whole rectangle is 4 by (2x + 3). Its two parts have areas 8x and 12.

Example 7: Distributing with a negative

Simplify −3(2x − 5) + 4x.

  1. Multiply each inside term by −3: −3 × 2x = −6x and −3 × (−5) = +15.
  2. So −6x + 15 + 4x.
  3. Combine like terms: −6x + 4x = −2x. Answer: −2x + 15.
Factoring 6x + 9: a rectangle with areas 6x and 9 has a shared height of 3, the greatest common factor, and widths 2x and 3, so 6x + 9 = 3(2x + 3) 6x + 9 = 3(2x + 3)6x2x933GCF of 6 and 9 is 3: the shared side
Factoring finds the side the two areas share. The largest shared side is the GCF, 3.

Example 8: Factoring out the GCF

Factor 12x + 18.

  1. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. GCF = 6.
  2. 12x ÷ 6 = 2x and 18 ÷ 6 = 3.
  3. Answer: 6(2x + 3). Check by distributing: 6 × 2x + 6 × 3 = 12x + 18. ✓

Equivalent expressions

Two expressions are equivalent if they give the same value for every value of the variable. 4(2x + 3) and 8x + 12 are equivalent; that’s why simplifying is allowed. A quick test: try a value. When x = 1, 4(2 + 3) = 20 and 8 + 12 = 20. One matching value doesn’t prove they are equivalent, but one mismatch proves they aren’t: 2(x + 3) and 2x + 3 give 8 and 5 when x = 1, so they are not equivalent.

How do we know?

The order of operations is a convention, an agreement, not a law of nature. 2 + 3 × 4 could mean 20 or 14; the agreement to multiply first makes it 14 for everybody. It also lets us write 3x + 5 without parentheses around 3x. When you want a different order, use parentheses: (2 + 3) × 4 = 20.

The distributive property is not a convention; it is always true, and area shows why. Three rows of (4 + 2) squares are 18 squares. Count the rows of 4 and the rows of 2 separately and you get 3 × 4 + 3 × 2 = 12 + 6 = 18. The squares didn’t change, only how we counted them. The same picture works for any lengths, including x.

A 3 by 6 grid of 18 squares split into a 3 by 4 part of 12 squares and a 3 by 2 part of 6 squares, showing 3 × (4 + 2) = 3 × 4 + 3 × 2 = 18 3 × (4 + 2) = 3 × 4 + 3 × 23 × 4 = 123 × 2 = 618 squares either way
The same 18 squares counted two ways: 3 × 6, or 3 × 4 plus 3 × 2.

History: letters and symbols

For a long time, algebra was written mostly in words. In 1591 the French mathematician François Viète published work that used letters for quantities, both known and unknown: vowels for the unknowns and consonants for the known numbers.

The habit of using letters near the end of the alphabet, like x, y and z, for unknowns, and letters near the start, like a, b and c, for known numbers, is credited to René Descartes in La Géométrie (1637). Other symbols arrived bit by bit: Robert Recorde’s “=” in 1557 (see Simple Equations), and the signs < and >, which first appear in a 1631 book by Thomas Harriot, published ten years after he died.

Did you know?

The < and > signs in Harriot’s book may not be his. MacTutor says they were introduced by the book’s editor, and that Harriot himself used different symbols.

Practice problems

Try each one, then open it to compare.

1. Evaluate 30 − 12 ÷ 3 × 2.

Division and multiplication left to right: 12 ÷ 3 = 4, then 4 × 2 = 8. Then 30 − 8 = 22.

2. Evaluate 2 × 3² + (8 − 5).

Parentheses: 8 − 5 = 3. Exponent: 3² = 9. Multiply: 2 × 9 = 18. Add: 18 + 3 = 21.

3. Evaluate 4a − b when a = −3 and b = −5.

4(−3) − (−5) = −12 + 5 = −7.

4. Write an expression for “the product of 6 and a number m, decreased by 9.”

6m − 9.

5. Simplify 7y − 4 + 2y + 10.

7y + 2y = 9y and −4 + 10 = 6, so 9y + 6.

6. Simplify 5(x − 2) + 3x.

Distribute: 5x − 10 + 3x. Combine: 8x − 10.

7. Factor 8x + 20.

GCF of 8 and 20 is 4: 4(2x + 5). Check: 4 × 2x + 4 × 5 = 8x + 20. ✓

8. Are 3(x + 2) and 3x + 2 equivalent? Explain.

No. 3(x + 2) = 3x + 6. Test x = 0: 3(0 + 2) = 6 but 3 × 0 + 2 = 2.

Game: Expression Lab

Ten problems. For Evaluate, type a whole number like -7. For Simplify, type an expression like 5x + 3 (spaces and order don’t matter). Press Enter. One point per correct answer. Your best score is saved only on this device. About 4 minutes.

Talk About It

  • A calculator gives 8 for 16 ÷ 4 × 2, and a friend says 2. Who is right, and why?
  • Why can you combine 3x and 5x but not 3x and 5y?
  • Draw an area model for 5(x + 2). How does it show both forms?
  • If the order of operations were “left to right, always,” what would 2 + 3 × 4 equal? What would we have to write differently?

Sources

Credits: All diagrams on this page are original SVG drawings made for Jugami. Names and prices in the examples are made up for practice.