Regents Review #2

Equations

What type of Equations do we need to solve?

1) Simple Equations2) Equations with Fractions3) Quadratic Equations4) Literal Equations (solving for another variable)5) Equations that help us solve word problems

Simple Equations½ (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!

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}2) 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}

16

Equations with Fractions

There are two ways to “clear” fractions in Equations…

1) Multiply both sides of the equation by the LCD (Least Common Denominator)

2) Cross Multiply (only works for proportions )

d

c

b

a

Equations with FractionsCross Multiplication (solving proportions)

53

2

1

1

x

x

x)1)(2()53(1 xxx

253

22532

2

xxx

xxxx

340 2 xx

13

0103

0)1)(3(

xx

xx

xxCheck for solutions that will make the denominator equal to zero.

dc

ba

x = {3,1}

Equations With FractionsMultiplying by the LCD

14

1

6

12

xx

)(14

1

6

12121212

xx

12)1(3)12(2 xx

7x

125

123324

x

xx

11

15.15.2

14

6

6

15

14

1

6

1)(2

77

CHECK

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

cxayy inforSolve

cxay

a

xc

a

ay

a

xcy

hoftermsinformulatheSolve

bhA2

1

bhA2

b

bh

b

A

2h

b

2A

xcay 21

bhA

Word Problems

We can use equations to solve many different types of word problems.When solving a word problem, remember to…

1)Define all unknowns (set up “Let” statements)2)Write an equation relating all unknowns3)Solve the equation4)Determine the value of all unknowns5)Answer the question (use appropriate units)

Word Problems

Find two consecutive integers whose sum is -35.x: 1st consecutive integer

x + 1: 2nd consecutive integer

18

362

3512

35)1(

x

x

x

xx

-18

-17 (-18 + 1)

Remember:

Consecutive integers count by 1’sEx: x, x+1, x+2, x+3….

Consecutive odd or even integers count by 2’sEx: x, x+2, x+4, x+6…

Word ProblemsTicket 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 concert3x: number of tickets sold for the Saturday concert2x: number of tickets sold for the Friday concert

2x + 2(3x) + 2(2x) = 2160

2x + 6x + 4x = 2160 12x = 2160 x = 180

180 tickets540 tickets 3(180)360 tickets 2(180)

There were 540 tickets sold for the Saturday night concert

Word Problems

A person has 23 coins made up of dimes and quarters worth $3.35. How many coins of each type are there?

0.10x + 0.25(23 – x) = 3.35

10x + 25(23 – x) = 335

10x + 575 – 25x = 335 -15x + 575 = 335 -15x = -240 x = 16

16 dimes 7 quarters (23 – 16)

Check

7(25) + 16(10) = 335 175 + 160 = 335 335 = 335

COIN QTY VALUE($)

Dimes x .10x

Quarters 23 – x .25(23 – x)

Word ProblemsThe area of a rectangle is 120 square inches. If the length is two more than the width, find the dimensions of the figure. Width: x

Length: x + 2

A = lw120 = (x + 2)(x)120 = x2 + 2x0 = x2 + 2x – 1200 = (x + 12)(x – 10)x + 12 = 0 x – 10 = 0 x = -12 x = 10 Reject

The width is 10 inchesThe length is 12 inches

Word ProblemsDistance, Rate, Time (D = RT)

Two Options

1)Finding Rate

2) Finding Time

RT

D

TR

D

D

TR

How do you know what to look for?

Distance: How far?Rate: How fast? Time: How long?

Word ProblemsHannah took a trip to her cousin’s house. She drove 120 miles to get there. If it took her 1.2 hours to get halfway to her cousin’s house, what was her average rate of speed?

Important Information: Distance traveled: 60 miles (120/2)Time: 1.2 hours Rate: ?

miles/hour50R hours

milesR

T

DR

2.1

60

If Hannah decided to travel 10 mph faster on the way home, how long would it take her?

hours2T mph

milesT

R

DT

60

120

Now it’s your turn to review on your own! Using the information presented today and your review

packet, complete the practice problems in the packet.

Regents Review #3 is

FRIDAY, May 17th BE THERE!!!!