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Solving Equations With Variables on Both Sides (VOBS) Medina 1 Steps to solve get the letter by itself variable 1. Simplify on side of the equation using distributive property then combining like terms. 2. Simplify the other side of equation using distributive property then combining like terms. 3. Get the variable on one side of equationMOVE VARIABLE FIRST!!!! To avoid negative move the smaller variable to the bigger variable 4. Use inverse operations (do/undo) You may not have to do all the steps to solve the equation “You know you are done solving when there is a positive 1 in front of the variable.” Medina 2 Solving Equations with Justification 2x 6 4x 3x 9 Original problem 6x 6 3x 9 Combine like terms To Solve: 3x 3x Subtraction Property of Equality Simplify 3x 6 9 D U •3 -6 +6 ÷3 6 6 3x 15 3 3 1x 5 Addition Property of Equality Simplify Division Property of Equality Check solution Medina 3 Solving Equations with Justification Check: x 5 2x 6 4x 3x 9 6x 6 3x 9 6 5 6 3 5 9 30 6 15 9 24 24 Medina 4 Solving Equations with Justification 4 x 3 2x 3x 6 Original problem 4x 12 2x 3x 6Distributive Property Combine like terms 12 3 x 6 6x To Solve: 3x Subtraction Property of 3x Equality D U Simplify 6 3 x 12 • 3 - 12 +12 ÷3 12 12 3x 18 3 3 1x 6 Subtraction Property of Equality Simplify Division Property of Equality Check answer Medina 5 Solving Equations with Justification 4 x 3 2x 3x 6 Check: x -6 4x 12 2x 3x 6 6x 12 3x 6 6 -6 12 3 - 6 6 -36 12 -18 6 24 24 Medina 6 Solving Equations with Justification 5𝑥 − 6 − 𝑥 = 2 𝑥 − 7 Original problem To Solve: Property 5𝑥 −6 + 𝑥 = 2 𝑥 − 7 Distributive on left side 5𝑥 − 6 + 𝑥 = 2𝑥 −14 Distributive Property on right side 6𝑥 −6 = 2𝑥 − 14 Combine like terms D U Subtraction Property −2𝑥 −2𝑥 of Equality •4 +6 4𝑥 −6 = −14 -6 ÷4 Simplify Property of + 6 + 6 Addition Equality 4𝑥 = −8 Simplify 4 4 Division Property of Equality Check answer 1 𝑥 = −2 Medina 7 Solving Equations with Justification 5𝑥 − 6 − 𝑥 = 2 𝑥 − 7 5 −2 − 6 − −2 Check: 𝑥 = −2 = 2 −2 − 7 −10 − 6 +2 = 2 −9 −10 − = −18 8 −18 = −18 Medina 8 Solving Equations with Justification 7𝑥 − 49 + 4𝑥 = −2 𝑥 + 5 Original problem To Solve: Combine like terms 11𝑥 −49 = −2 𝑥 + 5 on left side Distributive Property −2𝑥 −10 on right side 11𝑥 − 49 = +2𝑥 Addition Property of +2𝑥 Equality D U 13𝑥 −49 = −10 Simplify • 13 + 49 Addition Property of + 49 + 49 - 49 ÷ 13 Equality 13𝑥 = 39 Simplify 13 Division Property of Equality 13 Check answer 1𝑥 = 3 Medina 9 Solving Equations with Justification 7𝑥 − 49 + 4𝑥 = −2 𝑥 + 5 7 3 − 49 + 4 3 = −2 3 + 5 Check: 𝑥=3 21 −49 +12 = −2 8 −28 +12 = −16 −16 = −16 Medina 10 Determining the number of solutions: There are 3 possibilities No solution 1 solution Get the variable by itself and the variable equals a value x=-1 Infinitely Many Solutions The variables cancel out but the values don’t equal one another 8=10 There is only one value that can replace the variable that will make the statement (equation) true. There is NO value that can replace the variable, that will make the statement (equation) true. The variables cancel out and the values equal one another 10=10 Any value that will replace the variable will make the statement (equation) true. Medina 11 Solving Equations with VOBS D U •5 +9 2x 6 4x 3x 9 2x 6 3x 9 2x 2x 6 5x 9 -9 ÷5 1 solution 9 9 15 5x 5 5 3 1x Only -3 will make the statement true Medina 12 Solving Equations with VOBS Determining the number of solutions: 3x 3 5x 8x 9 8x 3 8x 9 8x 8x 3 9 No solution Ask yourself does 3 = 9? There is NO value of “x” that will make the statement true. Medina 13 Solving Equations with VOBS Determining the number of solutions: 5 x 3 4x 9x 15 5x 15 4x 9x 15 9x 15 9x 15 9x 9x -15 - 15 Infinite Solutions Ask yourself does -15 = -15? Any value of “x” will make the statement true. Medina 14