Equations Regents Review #2 Roslyn Middle School Research Honors Integrated Algebra

What type of Equations do we need to solve? Simple Equations Equations with Decimals Equations with Fractions Quadratic Equations Literal Equations (solving for another variable) Equations that help us solve word problems

Simple Equations ½ (2x – 10) = 5x – (6x + 9) x – 5 = 5x – 6x – 9 x – 5 = -x – 9 2x = -4 x = -2 Always check solution(s) to any equation ½ (2 -2 – 10) = 5(-2) – (6 -2 + 9) ½ (-14) = -10 – (-3) -7 = -7 It checks!

Equations with Decimals 0.54 – 0.07x = 0.2x 100[0.54 – 0.07x] = 100[0.2x] 54 – 7x = 20x 54 = 27x 2 = x Check 0.54 – 0.07(2) = 0.2(2) 0.54 – 0.14 = 0.4 0.4 = 0.4

Equations with Fractions There are two ways to “clear” fractions in Equations… Multiply both sides of the equation by the LCD (Least Common Denominator) 2) Cross Multiply (only works for proportions )

Equations With Fractions Multiplying by the LCD LCD: 6

Equations with Fractions Multiply by the LCD Quadratic Equation Check for extraneous solutions (solutions that will make any denominator equal to zero)

How do we solve Quadratic Equations? 1) x2 = a Example: x2 = 16 Take the square root of both sides x = x = 4 or x = {4,-4} x2 + bx + c = 0 Example: x2 – 5x + 6 = 0 Set all terms equal to zero (x – 2)(x – 3)= 0 Factor x – 2 = 0 x – 3 = 0 Set each factor equal to zero x = 2 x = 3 Solve x = {2,3}

Equations with Fractions Cross Multiplication (solving proportions) Check for extraneous solutions Extraneous Solution: values of x that will make the denominator(s) equal to zero

Literal Equations When solving literal equations, isolate the indicated variable using inverse operations

Word Problems We can use equations to solve many different types of word problems. When solving a word problem, remember to… Define all unknowns (set up “Let” statements) Write an equation relating all unknowns Solve the equation Determine the value of all unknowns Answer the question (use appropriate units)

Word Problems Ticket sales for a music concert totaled $2,160. Three times as many tickets were sold for the Saturday night concert than the Sunday afternoon concert. Two times as many tickets were sold for the Friday night concert than the Sunday afternoon concert. Tickets for all three concerts sold for $2 each. Find the number of tickets sold for the Saturday night concert. x: number of tickets sold for the Sunday concert 3x: number of tickets sold for the Saturday concert 2x: number of tickets sold for the Friday concert 180 tickets 540 tickets 3(180) 360 tickets 2(180) 2x + 2(3x) + 2(2x) = 2160 There were 540 tickets sold for the Saturday night concert 2x + 6x + 4x = 2160 12x = 2160 x = 180

Word Problems A person has 23 coins made up of dimes and quarters worth $3.35. How many coins of each type are there? 16 dimes 7 quarters (23 – 16) x: the number of dimes 23 – x: the number of quarters 0.10x + 0.25(23 – x) = 3.35 Check 7(25) + 16(10) = 335 175 + 160 = 335 335 = 335 10x + 25(23 – x) = 335 10x + 575 – 25x = 335 -15x + 575 = 335 -15x = -240 x = 16

Word Problems Ben is 6 years old Emma is 18 years old 3(6) Emma is three times as old as Ben. In 6 years, Ben will be one-half as old as Emma. What are their ages now? Present Age Future Age (in 6 years) Ben x x + 6 Emma 3x 3x + 6 x: Ben’s age now 3x: Emma’s age now Ben will be ½ as old as Emma x + 6 = ½ (3x + 6) Ben is 6 years old Emma is 18 years old 3(6) x + 6 = ½ (3x + 6) x + 6 = 1.5x + 3 -.5x = - 3 x = 6

Word Problems Distance, Rate, Time (D = RT) Two Options D1 = D2 2) D1 + D2 = Total Distance of 1 and 2

Word Problems D1 = D2 RT = RT 12x = 30(x – 3) 12x = 30x – 90 A cyclist leaves Bay Shore traveling at an average rate of 12 miles per hour. Three hours later, a car leaves Bay Shore, on the same route, traveling at an average speed of 30 miles per hour. How many hours will it take the car to catch up to the cyclist? 12 mph x: number of hours traveled by the cyclist x – 3: number of hours traveled by the car 5 hrs 2 hrs (5 – 3) 30 mph D1 = D2 RT = RT 12x = 30(x – 3) 12x = 30x – 90 -18x = -90 x = 5 The cyclist traveled 5 hours and the car traveled 2 hours. It took the car 2 hours to catch the cyclist.

Regents Review #2 Now it’s time to study Regents Review #2 Now it’s time to study! Using the information from this power point and your review packet, complete the practice problems.