Using the Addition and Multiplication Principles Together Section 2.3 Using the Addition and Multiplication Principles Together

Example Solve for w. 3w  9 = 24 3w  9 + 9 = 24 + 9 3w = 15 w = 5 Use the addition principle to add 9 to both sides of the equation. 3w  9 + 9 = 24 + 9 3w = 15 Use the division principle to divide both sides of the equation by 3. w = 5 Check your answer in the original equation. 3(5)  9 = 24 15  9 = 24  24 = 24

Example Solve for x. 9x = 6x + 15 9x + (6x) = 6x + (6x) + 15 3x = 15 Add –6x to both sides. Notice 6x + (–6x) eliminates the variable on the right side. 9x + (6x) = 6x + (6x) + 15 3x = 15 Use the division principle to divide both sides of the equation by 3. x = 5 Check your answer in the original equation. 9(5) = 6(5) + 15 45 = 30 + 15  45 = 45

Example  Solve for c. 9c + 5 = 3c  13 9c + 5 + (5) = 3c  13 + (5) Add 5 to both sides of the equation. 9c = 3c  18 9c + (3c) = 3c  18 + (3c) Add (3c) to both sides of the equation. 6c = 18 Divide both sides of the equation by 6. c = 3 9(3) + 5 = 3(3)  13 Check your answer in the original equation. 27 + 5 = 9  13 22 = 22  4

Example Solve for x. 5 + 7x  19 = 8x  6 + 3x 7x  14 = 11x  6 Add like terms on both sides of the equation. 7x  14 = 11x  6 Add 14 to both sides of the equation. 7x  14 + 14 = 11x  6 + 14 7x = 11x + 8 Add (11x) to both sides of the equation. 7x  11x = 11x + 8  11x  4x = 8 Divide both sides of the equation by 4. Be sure to check your answer! 5

Solving Equations with Parentheses The equations that you just solved are simpler versions of the equations that will now discuss. These equations contain parentheses. If the parentheses are first removed, the problem then becomes just like those encountered previously. We can use the distributive property to remove the parentheses.

Example Solve for d. 5d  6(d + 1) = 2d  6 5d  6d + (6) = 2d  6 Distribute to remove the parentheses.  d + (6) = 2d  6 Add like terms.  d + (6) + 6 = 2d  6 + 6 Add 6 to both sides of the equation.  d = 2d Simplify.  d  2d = 2d  2d Add (2d) to both sides of the equation.  3d = 0 Simplify. d = 0 Be sure to check your answer! 7

Example Solve for x. 0.3(1.2x – 3.6) = 4.2x – 16.44 Remove parentheses. 0.36x – 0.36x – 1.08 = 4.2x – 0.36x – 16.44 Subtract 0.36x from both sides. –1.08 = 3.84x – 16.44 Combine like terms. –1.08 + 16.44 = 3.84x – 16.44 + 16.44 Add 16.44 to both sides. 15.36 = 3.84x Simplify. Divide both sides by 3.84. 4 = x Be sure to check your answer! 8