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Solving Equations Recall that the solution to an equation is any value for the variable that makes the equation true. t = 4 is the solution of t + 2 = 6 Goal: apply the addition and multiplication principles to solve equations by isolating the variable. Examples: Verify that the given values are solutions to the equations. 1. x + 8 = 10 when x is 2 2. x + 8 = 6 when x is –2 3. 2y = 14 when y is 7 The equations: x = –2 and x + 8 = 6 are both true when x is –2. Equations that have the same solution are called equivalent equations. To solve an equation – 1. Use algebra principles. 2. Isolate the variable. The rules of algebra allow us to change an equation without changing the solution. 1 The Addition Principle: For real numbers a, b, and c, a = b is equivalent to a + c = b + c. Adding the same number to both sides of an equation does not change the solution. Examples: Use the addition principle to isolate the variable and solve. 1. x + 8 = 6 2. 9.5 = w – 1.1 The Multiplication Principle: For real numbers a, b, and c, with c ≠ 0, a = b is equivalent to a . c = b . c Multiplying both sides of an equation by the same, nonzero number does not change the solution. Note: The multiplication principle can act just like a division principle: For a, b, and c real numbers, c ≠ 0, if a = b, then a b 1 1 . = ⋅ a = ⋅ b or c c c c Example: Use the multiplication principle to isolate the variable and solve: 2x 1 = 3 4 Examples: Use the multiplication principle to isolate the variable and solve: 1. 2x = 14 2. 1.2t = 36 2 Examples: Choose the correct principle to isolate the variable and solve. 2 u 1. 7 = 3 2. 3 + v = 19 3. 1.4 = x – 0.5 3