Math A-100 IUSB Sec 2.1 detailed notes – Week 2 Determining if you have a solution A “solution” in math is simply a value that makes an equation true, i.e. a number that works There are two ways you can do this: 1) Plug the number into the equation in place of the variable and see if the math works out a. If it does work, then the number is “a solution” b. If it does not work out, then the number is NOT “a solution” 2) Work the problem out to see what the variable should be (i.e. solve the equation), then compare you answer to the number you are given a. If they match, then the number is “a solution” b. If they do not match, then the number is NOT “a solution” Example 1: is x = -3 a solution for 4x + 5x = -27 Option 1: Put -3 in for x: 4(-3) + 5(-3) = -27 -12 + -15  -27 -3 works, so it is a solution. Option 2: Solve for “x” 4x + 5x = -27 9x = -27  they match, so -3 is a solutuion x = -3 Example 2: is x = 4 a solution for 5x2 = 40 5(4)2 Option 1: Put 4 in for x: = 40 5(16) = 40 80 = 40 FALSE  4 does NOT work -4 is NOT a solution. Option 2: Solve for “x” 5x 2 = 40 x2 =8 take the square root of both sides x = sqrt(8)  does NOT match 4, so 4 is NOT a solution NOTE: you will be given several numbers to check for in each HW question. Check each of them individually – there canb be more than one solution! Example 3: 2 for 5x = 80 check to see if each is a solution: 4, 0, -4 Checking 4: 5(4)2 = 80 5(16) = 80 80 = 80 it works, so 4 is a solution. Checking 0: 2 5(0) = 80 5(0) = 80 0 = 80 it does NOT work, so 0 is NOT a solution. Checking -4: 5(-4)2 = 80 5(16) = 80 80 = 80 it works, so -4 is a solution. 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 Example 3: sqrt(x) = 8 x = 21 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 giving you: flip to multiplication & flip the fraction: x = 18 * 4 which gives you: 3 x = 24 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 Videos that may help: Solve Linear equations with Fractions One-step Equations with Fractions https://www.youtube.com/watch?v=COdQd39TTmU Two-step Equations with Fractions - clearing the fractions http://www.youtube.com/watch?v=uFptwRC6eXE&feature=youtu.be http://www.youtube.com/watch?v=m-lcy5KQN6I&feature=youtu.be http://www.youtube.com/watch?v=XFO71FBymWQ&feature=youtu.be