Solving Equations: The Addition and Multiplication Properties Section 3.2

Equation vs. Expression Statements like = 7 are called equations. An equation is of the form expression = expression An equation can be labeled as Equal sign left sideright side x + 5 = 9 2 Martin-Gay, Prealgebra, 5ed

Addition Property of Equality Let a, b, and c represent numbers. If a = b, then a + c = b + ca + c = b + ca + c = b + ca + c = b + cand a – c = b  c In other words, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation. 3 Martin-Gay, Prealgebra, 5ed

Multiplication Property of Equality Let a, b, and c represent numbers and let c  0. If a = b, then and a  c = b  c and In other words, both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution of the equation. 4 Martin-Gay, Prealgebra, 5ed

Solve for x. x  4 = 3 x  4 = 3 To solve the equation for x, we need to rewrite the equation in the form x = number. x = number. To do so, we add 4 to both sides of the equation. x  4 = 3 x  4 = 3 x  = Add 4 to both sides. x  = Add 4 to both sides. x = 7 Simplify. x = 7 Simplify. 5 Martin-Gay, Prealgebra, 5ed

Check x  4 = 3 Original equation x  4 = 3 Original equation 7  4 = 3 Replace x with 7. 7  4 = 3 Replace x with 7. 3 = 3 True. 3 = 3 True. Since 3 = 3 is a true statement, 7 is the solution of the equation. To check, replace x with 7 in the original equation. ? 6 Martin-Gay, Prealgebra, 5ed

Solve for x 4x = 8 4x = 8 To solve the equation for x, notice that 4 is multiplied by x. To get x alone, we divide both sides of the equation by 4 and then simplify. 1  x = 2 or x = 2 7 Martin-Gay, Prealgebra, 5ed

Check Check To check, replace x with 2 in the original equation. 4x = 8 Original equation 4x = 8 Original equation 4  2 = 8 Let x = 2. 4  2 = 8 Let x = 2. 8 = 8 True. 8 = 8 True. The solution is 2. ? 8 Martin-Gay, Prealgebra, 5ed

Using Both Properties to Solve Equations 2(2x – 3) = 10 Use the distributive property to simplify the left side. 4x – 6 = 10 Add 6 to both sides of the equation x = 4 9 4x – = x = 16 Divide both sides by 4.

Check Check To check, replace x with 4 in the original equation. Original equation 2(2x – 3) = 10 Original equation Let x = 4. 2(2 · 4 – 3) = 10 Let x = 4. 2(8 – 3) = 10 2(8 – 3) = 10 (2)5 = 10 True. (2)5 = 10 True. The solution is 4. ? 10 ? Martin-Gay, Prealgebra, 5ed