Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 How are expressions simplified by combining like terms? How are expressions simplified by combining like terms? How does the distributive property help to simplify expressions? How does the distributive property help to simplify expressions? Simplifying Expressions Essential Question

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 Vocabulary term like term coefficient constant equivalent expression

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 In the expression below, 7x, 5, 3y, and 2x are terms. A term can be a number, a variable, or a product of numbers and variables. Terms in an expression are separated by plus or minus signs.ConstantCoefficients

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 Like terms, such as 7x and 2x, can be grouped together because they have the same variable raised to the same power. Often, like terms have different coefficients. A coefficient is a number that is multiplied by a variable in an algebraic expression. A constant is a number that does not change. Constants, such as 4, 0.75, and 11, are also like terms. When you combine like terms, you change the way an expression looks but not the value of the expression. Equivalent expressions have the same value for all values of the variables.

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 Combine like terms. Additional Example 1: Combining Like Terms in One-Variable Expressions Identify like terms. Combine coefficients: 14 – 5 = 9 A. 14a – 5a 9a9a B. 7y + 8 – 3y – 1 + y Identify like terms; the coefficient of y is 1, because 1y = y. Combine coefficients: 7 – = 5 and 8 – 1 = 7 5y + 7

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 When you rearrange terms, move the operation in front of each term with that term. Helpful Hint

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 Combine like terms. Check It Out! Example 1 Identify like terms; the coefficient of q is 1, because 1q = q. Combine coefficients: 4 – 1 = 3 Identify like terms; the coefficient of c is 1, because 1c = c. Combine coefficients: 5 – 4 – 1 = 0 and 8 – 2 = 6 6 3q3q A. 4q – q B. 5c + 8 – 4c – 2 – c

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 Combine like terms. Additional Example 2A: Combining Like Terms in Two-Variables Expressions Identify like terms. Combine coefficients: 9 – 2 = 7 9x + 3y – 2x + 5 7x + 3y + 5 9x + 3y – 2x + 5

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 Combine like terms. Additional Example 2B: Combining Like Terms in Two-Variable Expressions Identify like terms. Combine coefficients: 5 – 2 = 3 and 7 – 3 = 4 5t + 7p – 3p – 2t 3t + 4p 5t + 7p – 3p – 2t

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 Combine like terms. Additional Example 2C: Combining Like Terms in Two-Variable Expressions No like terms. 4m + 9n – 2

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 Combine like terms. Check It Out! Example 2 Identify like terms. Combine coefficients: = 7 A. 2x + 5x – 4y + 3 7x – 4y + 3 2x + 5x – 4y + 3 Identify like terms. Combine coefficients: 9 – 4 = 5 and 7 – 2 = 5 B. 9d + 7c – 4d – 2c 5d + 5c 9d + 7c – 4d – 2c No like terms. C. 8g + c – 6 8g + c – 6

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 To simplify an expression, perform all possible operations, including combining like terms. You may need to use the Associative, Commutative, or Distributive Properties.

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 T HE D ISTRIBUTIVE P ROPERTY a(b + c) = ab + ac (b + c)a = ba + ca 2(x + 5)2(x) + 2(5)2x + 10 (x + 5)2(x)2 + (5)22x + 10 (1 + 5x)2(1)2 + (5x) x y(1 – y)y(1) – y(y)y – y 2 = = = = = = = = The product of a and (b + c):

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 (y – 5)(–2)= (y)(–2) + (–5)(–2) = –2y + 10 –(7 – 3x)= (–1)(7) + (–1)(–3x) = –7 + 3x = –3 – 3x (–3)(1 + x) = (–3)(1) + (–3)(x) Simplify. Distribute the –3. Simplify. Distribute the –2. Simplify. –a = –1 a Remember that a factor must multiply each term of an expression. Forgetting to distribute the negative sign when multiplying by a negative factor is a common error.

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 Simplify 6(5 + n) – 2n. Additional Example 3: Using the Distributive Property to Simplify Distributive Property Multiply. 6(5 + n) – 2n n – 2n 6(5) + 6(n) – 2n n Combine coefficients: 6 – 2 = 4. Identify like terms n – 2n

Simplifying Algebraic Expressions Evaluating Algebraic Expressions 3-2 Simplify 3(c + 7) – c. Check It Out! Example 3 Distributive Property Multiply. 3(c + 7) – c 3c + 21 – c 3(c) + 3(7) – c 2c + 21 Combine coefficients: 3 – 1 = 2. Identify like terms. 3c + 21 – c