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SOLVING EQUATIONS W/ FRACTIONS USING LCM! SWBAT: Solve Equations with Fractions. Fractions? Oh No! • Equations involving fractions can be quite difficult to work with. • There are easy ways to change the way a problem looks by using Multiplication and LCM’s! What is an LCM again? • It’s ok if you forgot… CLICK HERE TO WATCH THIS VIDEO!!! Ex 1: Solve the equation • Solve 3 7 a −1 = a + 9 11 11 "3 % "7 % 11$ a −1' = $ a + 9 '11 # 11 & # 11 & 3a −11 = 7a + 99 3a −11− 3a = 7a + 99 − 3a −11 = 4a + 99 −11− 99 = 4a + 99 − 99 −110 = 4a −27.5 = a Since both fractions have a denominator of 11, multiply each side by 11! Distribute the 11 to both sides! Now our equation is something more familiar! We can solve this! Lets undo operations! Ex 1B: Solve the equation • Solve 2x 4 = 3 3 ! 2x $ ! 4 $ 3# & = # & 3 " 3 % " 3% 2x = 4 Since both fractions have a denominator of 3, multiply each side by 3! Now our equation is something more familiar! We can solve this! Lets undo operations! x=2 Note: We could also have solved this problem by multiplying both sides by 3/2, the reciprocal of 2/3! Pt. 1 Practice… • 1. • 3. 3 7 x +1 = 4 4 2. 1 3 x + 3 = x −1 7 7 5 15 x=− 8 8 4. 9 1 2x + = 3+ x 4 4 1) x=1; 2) x=14 3) x= -3 4) x= 3/7 Ex 2: Different Denominators • Solve 2 1 1 − b+ = 3 9 18 " 2 1% " 1 % 18 $ − b + ' = $ '18 # 3 9 & # 18 & −12b + 2 =1 −12b = −1 −12b −1 = −12 −12 b= 1 12 3,9,and 18 have a LCM of 18, so multiply each side by 18! Distribute the 18 to both sides! Now our equation is something more familiar! We can solve this! Ex 2B: Different Denominators • Solve −3 2 = x 4 5 " 3% "2 % 20 $ − ' = $ x ' 20 # 4& #5 & −15 = 8x −15 8 = x 8 8 −15 =x 8 4 and 5 have a LCM of 20, so multiply each side by 20! Ex 2C: Different Denominators • Solve x +8 4+ x = 3 4 ! x +8$ ! 4 + x $ 12 # &=# &12 " 3 % " 4 % 4(x + 8) = (4 + x)3 4x + 32 =12 + 3x 4x + 32 − 3x =12 + 3x − 3x x + 32 =12 x + 32 − 32 =12 − 32 x = −20 3 and 4 have a LCM of 12, so multiply each side by 12! Pt. 2 Practice • 1. 1 7 x −8 = 2 8 2. 5 3 = x 8 16 • 3. 5 3 1 + x= 8 4 16 4. 2 3 4 x + = 1− x 5 7 7 x − 8 15 = 12 3 6. x − 7 2x + 3 = 4 2 • 5. 1) 71/4 2) 10/3 3) -3/4 4) 10/17 5) 68 6) -13/3 Ex 3: Distribution of FRACTIONS?? • Solve 2 ( x + 4) =16 3 2 (3) ( x + 4) =16(3) 3 2 ( x + 4) = 48 2x + 8 = 48 2x + 8 − 8 = 48 − 8 2x = 40 2x 40 = 2 2 x = 20 Distributing the 2/3 would give us more fractions… We can undo the fraction by multiplying by 3, then only have to distribute the 2! Ex 3: OR…… • Solve 2 ( x + 4) =16 3 ! 3$2 ! 3$ # & ( x + 4) =16 # & "2%3 "2% 1( x + 4) = 24 x + 4 = 24 x = 20 We can also undo the fraction by multiplying by its reciprocal! When multiplying a fraction by its Reciprocal, we always will get a product of 1! Pt. 3 Practice… 1 ( x + 4) =10 4 2) 3) 1 10x −10 = 5x −17 ( ) 4) 1) 5 1) 36, 2) 7, 3) 5, 4) 0 3 1 = (10 − x ) 2 2 2 1 (10x + 6) = (15x +12) 3 3 Ex 4: Variables in Denominators • Solve 2 −8 = 3 x ! 2 $ ! −8 $ 3x # & = # & 3x " 3% " x % 2x = −24 2x −24 = 2 2 x = −12 The Denominators 3 and x can be eliminated by multiplying both sides each one! Ex 4B: Variables in Denominators? • Solve x, 3x and 9 have a LCM of 9x... 2 4 2 − = Multiply by 9x! Clear those demoninators! x 3x 9 "2 4 % "2% 9x $ − ' = $ ' 9x # x 3x & # 9 & 18 −12 = 2x 6 = 2x 3= x Ex 4C: Variables in Denominators • Solve 9 −63 = x −1 28 " 9 % " −63 % ( x −1) (28)$# '& = $# '& ( x −1) (28) x −1 28 (28) ( 9) = (−63) ( x −1) 252 = −63x + 63 189 = −63x −3 = x The Denominators 28 and x – 1 can be eliminated by multiplying both sides each one! Pt. 4 Practice… 1) 4 −12 = 5 x 2) 2 −26 = x − 5 65 3) 3 12 − = 4 2x 4) 3 −18 = x + 4 −5x − 30 5) 5 17 +6 = x x 6) x + 2 x −1 + =5 3 6 1) -15 2) 0 3) -8 4) 6 5) 2 6) 9 Boom! • Solve 2 − x x + 3 −1 − = x 3x 3