Learn: two ways to count the same supplies
Each art kit contains x pencils and 3 erasers. Two kits contain 2(x + 3) items. Counting pencils and erasers separately gives 2x + 6 items. These expressions describe the same total for every allowed number of pencils.
Vocabulary before the steps
- A variable, such as x, represents a number. For the kit story, x is a nonnegative whole number.
- An expression combines numbers, variables and operations. It does not by itself assert that two quantities are equal.
- A term is a part separated by addition or subtraction; include its sign. In 3x − 2, the terms are 3x and −2.
- A coefficient multiplies the variable part. The coefficient of −4x is −4.
- Like terms have the same variable part, including powers. Constants are like terms with other constants.
Why can we combine 4x and 2x?
Four groups of x plus two groups of x make six groups of x: 4x + 2x = (4 + 2)x = 6x. But 4x and 2x² are not like terms. The expressions x and x² generally have different values, so 4x + 2x² cannot be combined into 6x².
Worked example: distribute a negative multiplier
Simplify 2(x + 3) − 4(x − 1).
- Distribute 2 to both terms: 2(x + 3) = 2x + 6.
- Treat the second multiplier as −4: −4(x − 1) = −4x + 4. A negative times a negative gives a positive.
- Combine the results: 2x + 6 − 4x + 4.
- Group like terms: (2 − 4)x + (6 + 4) = −2x + 10.
- Interpret: the shorter expression has the same value as the original for every real x. We have not found a particular value of x.
Explore: check a value, then explain the rule
Choose x to compare the two expressions numerically. A matching example can help catch an arithmetic error; it does not prove an identity for every x. The distributive property and combining like terms provide that proof.
At x = 2: 2(2 + 3) − 4(2 − 1) = 6; −2(2) + 10 = 6.
Another example: subtract a whole group
For 3x + 2 − (x + 5), the minus sign multiplies every term in the parentheses by −1. So 3x + 2 − x − 5 = 2x − 3. Writing 3x + 2 − x + 5 would change the expression.
Practice: show the intermediate line
1. Simplify 5x + 3 + 2x − 8.
(5 + 2)x + (3 − 8) = 7x − 5. Variable terms and constants are grouped separately.
2. Simplify −3(2x − 4) + x.
Distribute first: −6x + 12 + x. Then combine: −5x + 12.
3. Simplify 2x² + 3x + x² − x.
Combine matching powers: (2 + 1)x² + (3 − 1)x = 3x² + 2x. These remaining terms are not like terms.
4. A student says 2(x + 3) = 2x + 3. What was missed?
The 2 must multiply the 3 as well as x. The correct result is 2x + 6. At x = 0, the original is 6 but the incorrect expression is 3.
Review: preserve the value
Distribute to every term. Keep each sign with its term. Combine only matching variable parts. Simplifying rewrites an expression; solving an equation finds values that make a statement true. Continue with solving linear equations or pre-algebra.
Optional reference: OpenStax: the distributive property.