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Simplifying Linear Expressions: Like Terms, Distributing and Factoring

Combine like terms, use the distributive property, and factor out the greatest common factor.

Practice this skill 20 min Linear expressions: combine, factor, expand Leer en español
Key idea

What you'll learn

An algebraic expression is a math phrase with numbers, letters and operations, such as 3x + 12 or 5(2x - 3). In this lesson you will learn to:

  • combine like terms so a long expression becomes short,
  • use the distributive property to multiply (expand) an expression like 4(2x - 5),
  • factor out the greatest common factor (GCF) to go the other way, from 6x + 15 to 3(2x + 5),
  • add and subtract whole expressions, including the tricky case of subtracting something in parentheses.

Why it matters on the GED: almost every algebra question uses these moves. Sometimes the question asks you directly, "Which expression is equivalent to …?" More often, simplifying is the first step of solving an equation, writing a formula from a word problem, or finding a perimeter. If you can simplify quickly and without sign errors, the rest of the algebra section gets much easier.

In real life: a phone bill, a paycheck with overtime, or the cost of tickets plus fees can all be written as expressions. Simplifying shows the true cost per item and the fixed charges at a glance.

Words to know

  • Variable — a letter that stands for a number that can change (in 4x + 7, the variable is x).
  • Term — one piece of an expression separated by + or - signs (4x + 7 has two terms: 4x and 7).
  • Coefficient — the number multiplied by a variable (in 4x, the coefficient is 4; in x it is 1; in -x it is -1).
  • Constant — a term with no variable (the 7 in 4x + 7).
  • Like terms — terms with exactly the same variable part (3x and -8x are like terms; 3x and 3 are not).
  • Linear expression — an expression where the variable has no exponent bigger than 1 (2x - 9 is linear; x^2 + 1 is not).
  • Distributive property — a(b + c) = ab + ac: the number outside multiplies every term inside (2(x + 5) = 2x + 10).
  • Expand — remove parentheses by distributing (3(x - 1) expands to 3x - 3).
  • Factor — write as a product; the reverse of expanding (3x - 3 = 3(x - 1)).
  • Greatest common factor (GCF) — the largest number (and variable) that divides every term (12x + 18 has GCF 6).
  • Equivalent expressions — expressions that give the same value for every value of the variable (2(x + 3) and 2x + 6).

All Math words in the glossary

Combining like terms

You can only add or subtract like terms. Think of x as a label, like "boxes". Three boxes plus five boxes is eight boxes: 3x + 5x = 8x. But three boxes plus five loose dollars is not eight of anything: 3x + 5 stays 3x + 5.

Rule: to combine like terms, add or subtract the coefficients and keep the variable part the same.

  • 7x + 2x = 9x
  • 7x - 2x = 5x
  • x + 4x = 5x (remember x means 1x)
  • -3x + x = -2x
  • 6x - 6x = 0 (the x terms cancel)

Step-by-step method for a long expression:

  1. Treat the sign in front of each term as part of that term. In 5x + 3 - 2x + 7, the terms are +5x, +3, -2x, +7.
  2. Group the x-terms together and the constants together: (5x - 2x) + (3 + 7).
  3. Combine each group: 3x + 10.
Algebra tiles for 5x + 3 − 2x + 7. Five blue x-tiles and two red negative x-tiles: two blue and two red make zero pairs, leaving 3x. The constants are 3 unit tiles and 7 unit tiles, 10 in all. Result: 3x + 10.Simplify 5x + 3 − 2x + 7x-terms: +5x and −2xconstants: +3 and +7xxxxx−x−x2 zero pairs = 0left: 3x1113111111173 + 7 = 10Result: 3x + 10x-tile = x, small square = 1, red = negative

Moving terms around is allowed because addition can be done in any order, but each term must take its own sign with it. A common error is to move -2x and forget its minus sign.

ExpressionGroup like termsSimplified
4x + 9 + 3x - 2(4x + 3x) + (9 - 2)7x + 7
8 - x + 5x - 12(-x + 5x) + (8 - 12)4x - 4
0.5x + 3 - 1.5x(0.5x - 1.5x) + 3-x + 3
\tfrac{1}{2}x + \tfrac{1}{4}x\left(\tfrac{2}{4} + \tfrac{1}{4}\right)x\tfrac{3}{4}x

The usual way to write the answer is the x-term first and the constant last, but 10 + 3x is exactly the same as 3x + 10. On the GED, answer choices may be written in either order.

The distributive property: expanding

When a number sits right in front of parentheses, it multiplies every term inside:

a(b + c) = ab + ac \qquad a(b - c) = ab - ac
Area model for 3 times (x + 4): a rectangle 3 units tall and x + 4 units wide, split into a 3-by-x part with area 3x and a 3-by-4 part with area 12, so the total area is 3x + 12.x433x12Total area: 3(x + 4) = 3x + 12

The picture shows why. A rectangle 3 tall and x + 4 wide is made of two smaller rectangles: one is 3 by x (area 3x) and one is 3 by 4 (area 12). So 3(x + 4) = 3x + 12.

Examples:

  • 4(2x - 5) = 4 \cdot 2x - 4 \cdot 5 = 8x - 20
  • \tfrac{1}{2}(6x + 10) = 3x + 5
  • (x + 7) \cdot 3 = 3x + 21 (the number can be on the right, too)
Box model for 4(2x − 5): one row labelled 4 and two columns labelled 2x and −5. The cells are 4 · 2x = 8x and 4 · (−5) = −20 (shown in red as negative), so 4(2x − 5) = 8x − 20.4 · 2x = 8x4 · (−5) = −202x−544(2x − 5) = 8x − 20The 4 multiplies every term inside.

Negative numbers outside. Multiply the sign too. A negative times a negative is positive.

  • -2(x + 6) = -2x - 12
  • -2(x - 6) = -2x + 12 because (-2)(-6) = +12
  • -(3x - 8) = -3x + 8 — a minus sign alone in front of parentheses means -1, so every sign inside flips.

Subtracting a group. In 3(2x - 4) - 2(x - 5), the number outside the second group is -2, not 2. So:

3(2x - 4) - 2(x - 5) = 6x - 12 - 2x + 10 = 4x - 2

Always distribute first, then combine like terms. This order follows the order of operations: multiplication (the distributing) comes before addition and subtraction.

Factoring out the GCF: going backwards

Factoring undoes distributing. You look for the largest number that divides every term, pull it out in front, and write what is left inside the parentheses.

Steps to factor out the GCF:

  1. Find the GCF of the coefficients and constants. For 12x + 18: factors of 12 are 1, 2, 3, 4, 6, 12; factors of 18 are 1, 2, 3, 6, 9, 18. The greatest one in both lists is 6.
  2. Divide each term by the GCF: 12x \div 6 = 2x and 18 \div 6 = 3.
  3. Write the GCF outside and the results inside: 12x + 18 = 6(2x + 3).
  4. Check by distributing back: 6(2x + 3) = 12x + 18 ✓.
Factoring 12x + 18 with an area model: a rectangle with total area 12x + 18 is split into parts of area 12x and 18. The common height is the GCF 6, so the widths are 12x ÷ 6 = 2x and 18 ÷ 6 = 3. Therefore 12x + 18 = 6(2x + 3).12x182x36(GCF)12x ÷ 6 = 2x18 ÷ 6 = 312x + 18 = 6(2x + 3)

More examples:

  • 6x + 15 = 3(2x + 5)
  • 8x - 20 = 4(2x - 5)
  • 7x + 7 = 7(x + 1) — when a term is the GCF itself, a 1 is left behind, not 0.
  • -10x + 25 = -5(2x - 5) — you may pull out a negative GCF; then every sign inside flips.
  • 9x + 6y - 3 = 3(3x + 2y - 1) — it works with more than two terms.

"Completely" factored means the parentheses have no common factor left. 24x + 36 = 2(12x + 18) is equivalent, but it is not completely factored because 12x + 18 still shares a 6. The GCF is 12: 24x + 36 = 12(2x + 3).

Can you pull out x? Only if every term has an x. In 6x + 15, the 15 has no x, so x is not a common factor.

Formula

Key properties (memorize — not on the formula sheet)

a(b + c) = ab + ac \qquad\qquad a(b - c) = ab - ac -(b - c) = -b + c \qquad\qquad ax + bx = (a + b)x
  • a is the number (or term) outside the parentheses; b and c are the terms inside.
  • The first line is the distributive property. Read left to right, it expands. Read right to left, it factors.
  • The last rule, ax + bx = (a + b)x, is why combining like terms works: 3x + 5x = (3 + 5)x = 8x.

These rules are not on the GED formula sheet. They are basic properties of numbers that you are expected to know.

Worked example

Example 1 (easy): Combine like terms

Simplify 7x + 4 - 3x + 9.

  1. List the terms with their signs: +7x, +4, -3x, +9.
  2. Group: (7x - 3x) + (4 + 9).
  3. Combine: 7x - 3x = 4x and 4 + 9 = 13.

The figure sums it up:

Sorting the terms of 7x + 4 − 3x + 9: the chips +7x and −3x go into the x-terms bin, giving 7x − 3x = 4x; the chips +4 and +9 go into the constants bin, giving 13. Answer: 4x + 13.7x + 4 − 3x + 9+7x+4−3x+9x-terms7x − 3x = 4xconstants4 + 9 = 13Answer: 4x + 13

Answer: 4x + 13

Check: let x = 1. Original: 7 + 4 - 3 + 9 = 17. Answer: 4 + 13 = 17 ✓

Worked example

Example 2: Distribute with a negative

Simplify 5(2x - 3) - 4(x - 6).

  1. Distribute 5: 5 \cdot 2x - 5 \cdot 3 = 10x - 15.
  2. Distribute -4 (keep the minus sign with the 4): (-4)(x) + (-4)(-6) = -4x + 24.
  3. Write it all out: 10x - 15 - 4x + 24.
  4. Combine: (10x - 4x) + (-15 + 24) = 6x + 9.

The figure sums it up:

Two box models for 5(2x − 3) − 4(x − 6). First box: row 5, columns 2x and −3, cells 10x and −15. Second box: row −4, columns x and −6, cells −4x and +24 (negative times negative is positive). Adding all cells: 10x − 15 − 4x + 24 = 6x + 9.10x−152x−35−4x+24x−6−45(2x − 3)− 4(x − 6)+10x − 15 − 4x + 24= (10x − 4x) + (−15 + 24) = 6x + 9

Answer: 6x + 9

Check: let x = 1. Original: 5(-1) - 4(-5) = -5 + 20 = 15. Answer: 6 + 9 = 15 ✓

Worked example

Example 3 (word problem): Tickets and fees

Three friends each buy a concert ticket that costs x dollars plus a $6 service fee per ticket. They also pay one $4 parking charge for the car. Write a simplified expression for the total cost.

  1. Cost of one ticket with its fee: x + 6.
  2. Three tickets: 3(x + 6).
  3. Add parking: 3(x + 6) + 4.
  4. Distribute: 3x + 18 + 4.
  5. Combine constants: 3x + 22.

The figure sums it up:

Tape diagram for the ticket problem: three rows, one per friend, each made of a ticket bar x and a fee bar 6, bracketed as 3(x + 6), plus one parking bar of 4 for the car. Total: 3(x + 6) + 4 = 3x + 18 + 4 = 3x + 22 dollars.Three tickets, each x dollars + $6 fee, plus one $4 parking chargeFriend 1ticket xfee 6Friend 2ticket xfee 6Friend 3ticket xfee 63(x + 6)Car4← parking, paid onceTotal: 3(x + 6) + 4 = 3x + 18 + 4 = 3x + 22

Answer: 3x + 22 dollars

Check: if a ticket costs $50, each friend pays $56, so three pay $168, plus $4 parking is $172. Formula: 3(50) + 22 = 172 ✓

Worked example

Example 4 (word problem): Factor the GCF

A fence has a total length of 12x + 18 feet. It is built from 6 equal sections. Write an expression for the length of one section.

  1. Find the GCF of 12x and 18: it is 6.
  2. Factor: 12x + 18 = 6(2x + 3).
  3. Six equal sections, each 2x + 3 feet long.

The figure sums it up:

A fence of total length 12x + 18 feet drawn as 6 equal sections between posts, each section labelled 2x + 3. Because 12x + 18 = 6(2x + 3), one section is 2x + 3 feet long.Whole fence: 12x + 18 feet2x + 3#12x + 3#22x + 3#32x + 3#42x + 3#52x + 3#66 equal sections: 12x + 18 = 6(2x + 3)One section = 2x + 3 feet

Answer: 2x + 3 feet

Check: 6(2x + 3) = 12x + 18 ✓

Worked example

Example 5 (GED-hard): Find the missing number

For what value of a is a(3x - 2) + 4x equivalent to 19x - 10?

  1. Distribute: a(3x - 2) = 3ax - 2a.
  2. The whole expression is 3ax - 2a + 4x = (3a + 4)x - 2a.
  3. Match the x-coefficients: 3a + 4 = 19, so 3a = 15 and a = 5.
  4. Match the constants to confirm: -2a = -2(5) = -10 ✓

The figure sums it up:

Matching parts of equivalent expressions: the top row (3a + 4)x − 2a is matched with the bottom row 19x − 10. Matching the x-coefficients gives 3a + 4 = 19, so a = 5; matching the constants gives 2a = 10, so a = 5 again, which confirms the answer.a(3x − 2) + 4x = 3ax − 2a + 4x(3a + 4)x−2a19x−10matchmatch3a + 4 = 19 → a = 52a = 10 → a = 5 ✓

Answer: a = 5

Check: 5(3x - 2) + 4x = 15x - 10 + 4x = 19x - 10 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

Simplifying expressions is mostly mental math — you do not need the calculator to combine 7x and -3x. But the calculator is a great checker.

The "plug in a number" check: pick an easy value such as x = 2 and compute both the original expression and your answer. They must match.

For example, to check 5(2x - 3) - 4(x - 6) = 6x + 9 with x = 2:

  • Type 5 ( 2 × 2 − 3 ) − 4 ( 2 − 6 ) enter → 21.
  • Type 6 × 2 + 9 enter → 21 ✓

Use the (−) key (not the minus key) to enter a negative number like -3. Avoid x = 0 or x = 1 as your only test: they can hide mistakes. If two answer choices give the same value for one test number, try a second number.

Common mistake

Common mistakes

  • Distributing to the first term only. Wrong: 3(x + 4) = 3x + 4. Right: 3(x + 4) = 3x + 12 — the 3 multiplies both terms.
  • Losing the negative when distributing. Wrong: -2(x - 5) = -2x - 10. Right: (-2)(-5) = +10, so -2(x - 5) = -2x + 10.
  • Combining unlike terms. Wrong: 5x + 3 = 8x. Right: 5x and 3 are not like terms; 5x + 3 is already simplified.
  • Dropping a term's sign when regrouping. Wrong: 8 - x + 5x gives 6x + 8. Right: the first x-term is -x, so -x + 5x = 4x, giving 4x + 8.
  • Factoring out a number that is not the greatest. 24x + 36 = 2(12x + 18) is equivalent but not completely factored. Right: 12(2x + 3).
  • Leaving 0 instead of 1 when factoring. Wrong: 7x + 7 = 7(x + 0). Right: 7 \div 7 = 1, so 7(x + 1).
Test strategy

GED test strategy

  • Read what form they want. "Equivalent to" means any correct form. "Factored completely" means the GCF must be pulled out. "Simplified" means no parentheses and no like terms left.
  • Plug in a test number. If you are unsure, choose x = 2 (or x = 3). Find the value of the original expression, then test each answer choice. Only the equivalent one matches. This is often faster than doing the algebra.
  • Eliminate by the constant first. In "simplify 3(2x - 4) - 2(x - 5)", the constants are -12 and +10, so the constant must be -2. Cross out every choice without -2.
  • Watch the subtraction signs. Before distributing, circle any minus sign in front of parentheses. Many wrong answer choices come from that one step.
  • Check factoring by expanding. It takes five seconds to multiply back out, and it catches nearly every factoring error.
  • Keep moving. These questions should take about a minute. If you are stuck, plug in a number, pick the match, and go on.

Check yourself

Linear expressions: combine, factor, expand

Which expression is equivalent to 5x + 3 - 2x + 7?

Answer choices
Hint

Group the x-terms together and the plain numbers together. Keep each term's own sign.

Check yourself

Linear expressions: combine, factor, expandFill in

Use the distributive property to expand 4(2x - 5).

4(2x - 5) = x \; -

Type your answer in the box. Fractions like 3/4 are fine.

Hint

The 4 outside multiplies each term inside the parentheses, not just the first one.

Summary

  • Only like terms (same variable part) can be combined: add or subtract their coefficients. 3x + 5x = 8x, but 3x + 5 stays as it is.
  • Distribute the outside number to every term inside the parentheses: a(b + c) = ab + ac.
  • A minus sign in front of parentheses flips every sign inside: -(3x - 8) = -3x + 8.
  • Distribute first, then combine like terms.
  • Factor by pulling out the greatest common factor and dividing each term by it; check by expanding back.
  • Test any answer by plugging in a number such as x = 2 — equivalent expressions always give the same value.
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