Solving Linear Equations 12/9/14-12/10/14

Vocabulary Inverse operations: opposite operations that undo each other. Addition and subtraction are inverse operations. Multiplication and division are inverse operations. Combine Like Terms: mathematical process used to simplify an expression or to add or subtract polynomials. To do this, terms must have the same variable and exponent. Examples of LIKE TERMS: -8 and 27 5t and -9t 7x2 and 6x2 Isolating the Variable: Using inverse operations to undo addition, subtraction, multiplication, and division to get the variable alone.

Linear Equations Goal: The goal of solving a linear equation is to find the value of the variable that will make the statement (equation) true. Method: Perform operations to both sides of the equation in order to isolate the variable. A solution of a linear equation is a real number which, when substituted for the variable in the equation, makes the equation true.

Addition and Subtraction Properties of Equality Let a, b, and c represent algebraic expressions. 1. Addition property of equality: If a=b, then a + c = b + c 2. Subtraction property of equality: then a - c = b - c

Multiplication and Division Properties of Equality Let a, b, and c represent algebraic expressions. 1. Multiplication property of equality: If a=b, then ac = bc 2. Division property of equality: then 𝑎 𝑐 = 𝑏 𝑐 c ≠ 0

Steps for Solving a Linear Equation in One Variable Simplify/Combine Like Terms on both sides of the equation. Undo the Addition or Subtraction by using inverse operations. Use the addition or subtraction properties of equality to collect the variable terms on one side of the equation and the constant terms on the other. Undo the Multiplication or Division by using inverse operations. Use the multiplication or division properties of equality to make the coefficient of the variable term equal to 1. Check your answer by substituting your solution into the original equation.

Examples:

Examples (cont’d):

Distributive Property The Distributive Property of Multiplication is the property that states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. The Distributive Property says that if a, b, and c are real numbers, then: a(b + c) = (a x b) + (a x c) EXAMPLES:

Solving Algebraically and Explanations -11 – 9 = -11 + (-9) x = -20 Explanation: Original Equation Use Inverse Operations – subtract 9 from both sides and combine like terms. Use rules subtracting integers to simplify. Solution.

Check your SOLUTION for: x + 9 = -11 Substitute the –20 for x and evaluate using the correct order of operations. -20 + 9 Use rules for adding integers with opposite signs. In this case, subtract the absolute values. |-20| = 20 |9| = 9 20 - 9 = 11 Place the sign in your answer of the number with the greater absolute value. -20 has the greater absolute value, so… -20 + 9 = -11 The solution, -20, is the correct solution!

Solving Algebraically and Explanations 3 3 x = 1 Explanation: Original Equation Use Inverse Operations. Subtract 4 from both sides and combine like terms. Use Inverse Operations. Divide both sides by 3. Simplify to find solution.

Check your SOLUTION for: 3x + 4 = 7 Substitute the 14 3 for x and evaluate using the correct order of operations.

Solving Algebraically and Explanations Original Equation Distributive Property and Simplify. Inverse Operations. Add 6 to both sides and combine like terms. Use Inverse Operations. Divide both sides by 3. Simplify to find solution. Algebraic: 3(x - 2) = 8 3(x) – 3(2) = 3x – 6 = 8 3x – 6 + 6= 8 + 6 3x = 14 3 3 x = 14 3

Check your SOLUTION for: 3(x - 2) = 8 Substitute the 1 for x and evaluate using the correct order of operations.

Practice pg. 122 – TRY THESE A