Math A100 IUSB Sec. 2.3 – 2.5 detailed notes – Week 3 More on Solving Equations Recall from Week 2 that we had 3 options for solving equations with fractions: 1. Undo what they gave you 2. Make it into a proportion 3. “Clear the Fractions”. 1. Undo what they gave you: Example 1: 3x = 18 4 They gave you “3/4 multiplied by x”. Use division to undo the multiplication, i.e. divide both sides by 3/4 3x divide by 3 = 18 divided by 3 4 4 4 x = 18 divided by 3 4 x = 18 * 4 3 giving you: flip to multiplication & flip the fraction: which gives you: x = 24 2. Make it is into a proportion Example 1: 3x = 18 4 Turn it into 3x = 18 4 1 then cross multiply: (3x)(1) = 18*4  3x = 72  x = 24 3. “Clear the Fractions”: a. Get the bottom numbers the same for ALL numbers (including any whole numbers) b. Cross out ALL the bottom numbers and re-write the equation  Fractions are GONE. The equations looks like a normal equation now. Example 1: 3+x=6- x 2 5 4 Turn it into _3 + x = 6 - x 2 5 1 4 Make the bottoms the same  smallest number 2, 4, & 5 all go into is 20 3 + x = 6 – x becomes 2 5 1 4 30 + 4x = 120 – 5x 20 20 20 20 Cross out the bottoms and re-write: 30 + 4x = 120 – 5x becomes 30 + 4x = 120 – 5x 20 20 20 20 Solve like normal: 30 + 4x = 120 – 5x  9x = 90  x = 10  final answer Special Cases of Solving Equations There are two special cases when solving equations: 1) When every number you put in is a solution – if every number works, the answer is “All Real Numbers” (a.k.a. “infinitely many solutions”). These type of questions are also called Identities, because one side of the equation is the same, i.e. identical, to the other side. 2) When no number you put in is a solution - if no number works, the answer in “No Solution”. These type of questions are also called Contradictions, because no value ever works. 3) When you have just one solution, i.e. answer, these type of questions are also called Conditional Solutions, because they work “on the condition” that you are using the one value that works for the variable. NOTE: You do not have to memorize the words Identities, Contradictions, & Conditional Solutions, but I WILL expect you know All Real Solutions & No Solution Example 1: 3x + 5x = 8x 3x + 5x = 8x 8x = 8x x=x put your like terms together to get divide by 8 to undo the multiplying by 8 to get since x always equals itself  all numbers work  All Real Numbers 3x + 5x = 8x 8x = 8x 0=0 put your like terms together to get subtract 8x to undo the positive 8x to get 0 always equals itself  all numbers work  All Real Numbers or Example 2: 6x - 3(2x – 6) = 18 6x - 3(2x – 6) = 18 distribute to break up the ( ), giving you 6x - 6x + 18 = 18 18 = 18 Example 3: 6x - 3(2x – 5) = 18 6x - 3(2x – 5) = 18 6x - 6x + 15 = 18 15 = 18 Example 4: put your like terms together to get always true  all numbers work  All Real Numbers distribute to break up the ( ), giving you put your like terms together to get NOT true  No Solution 6x - (2x – 5) = - (7 – 3x) + x 6x - (2x – 5) = - (7 – 3x) + x 6x - 2x + 5 = - 7 + 3x + x 4x + 5 = -7 + 4x 5 = -7 Example 5: 6x - (2x + 5) = -5(1 – x) - x 6x - (2x + 5) = -5(1 – x) - x 6x - 2x - 5 = -5 + 5x - x 4x - 5 = -5 + 4x - 5 = -5 Example 6: distribute to break up the ( ), giving you put your like terms together to get subtract 4x to undo the positive 4x to get NOT true  No Solution distribute to break up the ( ), giving you put your like terms together to get subtract 4x to undo the positive x to get always true  All Real Numbers 8x - (2x + 5) = -5(3 – x) - x 8x - (2x + 5) = -5(3 – x) - x 8x - 2x - 5 = -15 + 5x - x 6x - 5 = -15 + 4x 2x - 5 = -15 2x = -10 x = -5 distribute to break up the ( ), giving you put your like terms together to get subtract 4x to undo the positive 4x to get add 5 to undo the subtract 5, giving you divide by 2 to undo the multiplying by 2 to get One Solution: x = -5 Solving a Formula for One Variable http://www.khanacademy.org/math/algebra/solving-linear-equations-andinequalities/solving_for_variable/v/rearrange-formulas-to-isolate-specific-variables http://www.khanacademy.org/math/algebra/solving-linear-equations-andinequalities/solving_for_variable/v/solving-for-a-variable http://www.khanacademy.org/math/algebra/solving-linear-equations-andinequalities/solving_for_variable/v/solving-for-a-variable-2