Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 5: Equations and Inequalities

Lesson 1: Expressions

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra (AMC 8), students learn to identify and work with mathematical expressions, including key vocabulary such as terms, coefficients, and constant terms. The lesson covers how to determine whether two expressions are equivalent and how to simplify expressions by combining like terms using the distributive property. Students practice writing and simplifying variable expressions through real-world word problems involving multiplication and addition of terms.

Section 1

Terms, coefficients, and like terms

Property

A term is a constant or the product of a constant and one or more variables. The constant that multiplies the variable(s) in a term is called the coefficient. Like terms are terms that are either constants or have the same variables with the same exponents.

Examples

  • In the expression 4x2+7y−34x^2 + 7y - 3, the terms are 4x24x^2, 7y7y, and −3-3. The coefficient of 4x24x^2 is 44, and the coefficient of 7y7y is 77.
  • Identify like terms in the list: 2a,5b2,6,9a,3b22a, 5b^2, 6, 9a, 3b^2. The like terms are 2a2a and 9a9a (both have the variable aa) and 5b25b^2 and 3b23b^2 (both have b2b^2).

Section 2

Testing Equivalence with Substitution

Property

Two algebraic expressions are equivalent if they name the same number for all values of the variable.

Examples

  • The expressions 4x+2x4x + 2x and 6x6x are equivalent. If we test x=3x=3, we get 4(3)+2(3)=12+6=184(3) + 2(3) = 12 + 6 = 18, and 6(3)=186(3) = 18.
  • The expressions 10−2x10-2x and 8x8x are not equivalent. If we test x=1x=1, we get 10−2(1)=810-2(1)=8 and 8(1)=88(1)=8. But if we test x=2x=2, we get 10−2(2)=610-2(2)=6 and 8(2)=168(2)=16. Since they are not equal for all values, they are not equivalent.
  • The expressions 9y−3y9y - 3y and 6y6y are equivalent. For any value of yy, subtracting three yy's from nine yy's always results in six yy's.

Explanation

Think of equivalent expressions as two different ways to write the same value. No matter what number you substitute for the variable, they will always produce the same result because they are mathematically identical.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Terms, coefficients, and like terms

Property

A term is a constant or the product of a constant and one or more variables. The constant that multiplies the variable(s) in a term is called the coefficient. Like terms are terms that are either constants or have the same variables with the same exponents.

Examples

  • In the expression 4x2+7y−34x^2 + 7y - 3, the terms are 4x24x^2, 7y7y, and −3-3. The coefficient of 4x24x^2 is 44, and the coefficient of 7y7y is 77.
  • Identify like terms in the list: 2a,5b2,6,9a,3b22a, 5b^2, 6, 9a, 3b^2. The like terms are 2a2a and 9a9a (both have the variable aa) and 5b25b^2 and 3b23b^2 (both have b2b^2).

Section 2

Testing Equivalence with Substitution

Property

Two algebraic expressions are equivalent if they name the same number for all values of the variable.

Examples

  • The expressions 4x+2x4x + 2x and 6x6x are equivalent. If we test x=3x=3, we get 4(3)+2(3)=12+6=184(3) + 2(3) = 12 + 6 = 18, and 6(3)=186(3) = 18.
  • The expressions 10−2x10-2x and 8x8x are not equivalent. If we test x=1x=1, we get 10−2(1)=810-2(1)=8 and 8(1)=88(1)=8. But if we test x=2x=2, we get 10−2(2)=610-2(2)=6 and 8(2)=168(2)=16. Since they are not equal for all values, they are not equivalent.
  • The expressions 9y−3y9y - 3y and 6y6y are equivalent. For any value of yy, subtracting three yy's from nine yy's always results in six yy's.

Explanation

Think of equivalent expressions as two different ways to write the same value. No matter what number you substitute for the variable, they will always produce the same result because they are mathematically identical.