Math A-100 IUSB Sec 2.2 detailed notes – Week 2 Solving Equations You will get a wide variety of different looking equations and ask you to solve for the variable, usually “x”. My best suggestion is to keep the following phrase in mind: Undo what they gave you: Example 1: 3x = 18 They gave you “3 multiplied by x” so use division to undo the multiplication, i.e. divide both sides by 3 3x/3 = 18/3 x=6 Example 2: x - 3 = 18 They gave you “x subtract 3” so use addition to undo the subtraction, i.e. add 3 to both sides: x – 3 + 3 = 18 + 3 x = 21 Example 3: sqrt(x) = 8 They gave you “square root of x” so square both sides to undo the square root: [Sqrt(x)]2 = [8]2 x = 64 Example 4: x2 = 49 They gave you “x squared so take the square root of both sides to undo the square: Sqrt(x2) = sqrt[49] Example 5: x=7 3x = 18 4 They gave you “3/4 multiplied by x”. When working with fractions, you have a few options: a) 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 x = 24 giving you: flip to multiplication & flip the fraction: which gives you: b) since you end up flipping the fraction and multiplying anyway, just go there directly and skip the first part: 3x * 4 = 18 * 4 4 3 3 x = 18 * 4 3 x = 24 giving you: which gives you: c) break it up and do it in parts: 3x = 18 4 They gave you “3 multiplied by x” & “x divided by 4”, so undo each one separately Multiply by 4 to undo the dividing by 4 3x * 4 = 18 * 4 4 which gives you: 3x = 72 then divide both sides by 3 to undo the multiplying: 3x/3 = 72/3 which gives you x = 24 Example 6: .4x = 18 Decimals are just like whole numbers. they gave you “.4 multiplied by x” so use division to undo the multiplication, i.e. divide both sides by .4 .4x/.4 = 18/.4 x = 45 Other strategies to consider: 1) Proportions – make it into a proportion and: a. Cross multiply if the bottoms of the fractions are different b. Clear the fractions if the bottoms of the fractions are the same Option a. Cross multiply if the bottoms of the fractions are different Example 1: 3x = 18 4 Turn it into 3x = 18 4 1 then cross multiply: (3x)(1) = 18*4  3x = 72 Example 2:  x = 24 3 = 18 4x Turn it into _3_ = 18 4x 1 then cross multiply: (3)(1) = 18(4 x)  3 = 72x Example 3: 3 + x = 18 4  x = 3/72 = 1/24 Turn it into _3 + x_ = 18 4 1 then cross multiply: (3 + x))(1) = 18(4 )  3 + x = 72  x = 69 Example 4: 3 + x = 18 - x 4 Turn it into _3 + x_ = 18 - x 4 1 then cross multiply: (3 + x))(1) = (4 )(18 – x)  3 + x = 72 – 4x  3 + 5x = 72  5x = 69  x = 69/5 Option b. Clear the fractions if the bottoms of the fractions are the same “Clearing the Fractions is a technique that a lot of students like, and GREATLY reduces their stress level!! To “Clear the Fractions”: 1) Get the bottom numbers the same for ALL numbers (including any whole numbers) 2) Cross out ALL the bottom numbers and re-write the equation  Fractions are GONE. The equations looks like a normal equation now. Revisiting Example 1 (above): 3x = 18 4 Re-write with 18 as a fraction, turning it into Make the bottoms the same: 3x = 18 4 1 3x = 18 4 1 becomes Cross out the bottoms and re-write: 3x = 72 4 4 3x = 72 4 4 becomes 3x = 72 Solve like normal: 3x = 72  x = 24  same answer as above Revisiting Example 2 (above): Turn it into 3 = 18 4x _3_ = 18 4x 1 Make the bottoms the same: 3 = 18 4x 1 becomes Cross out the bottoms and re-write: 3 = 72x 4x 4x 3 = 72x 4x 4x becomes 3 = 72x Solve like normal: 3 = 72x  x = 1/24  same answer as above Revisiting Example 3 (above): 3 + x = 18 4 Turn it into _3 + x_ = 18 4 1 Make the bottoms the same: 3 + x = 18 4 1 becomes 3 + x = 72 4 4 Cross out the bottoms and re-write: 3 + x = 72 becomes 3 + x = 72 4 4 Solve like normal: 3 + x = 72  x = 69  same answer as above Example 4: 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 Videos that may help: Proportions http://www.youtube.com/watch?v=j-va9qeYyOI http://www.youtube.com/watch?v=nWGwKmuBE1U