Chapter 7 Notes

7-1 Solving 2-Step Equations eTo solve a 2-step equation, first undo addition or subtraction. Then undo multiplication or division.

Examples 3n - 6 = n = 21 3 N = 7

Examples e15x + 3 = 48 eR/ = (-6) eB/ = 11 e9g + 11 = 2

Examples - Answers e15x + 3 = 48 x=3 eR/ = (-6) R=16 eB/ = 11 B= -6 e9g + 11 = 2 g= -1

Negative Coefficients eExamples: e5 - x = 17 e-a + 6 = 8 e-9 - y/7 = (-12) e13 - 6f = 31

Negative Coefficients - Answers eExamples: e5 - x = 17 x=(-12) e-a + 6 = 8 a=(-2) e-9 - y/7 = (-12) y=(21) e13 - 6f = 31 f=(-3)

Word Problems eLynne wants to save $900 to go to Puerto Rico. She saves $45 each week and now has $180. To find how many more weeks w it will take to have $900, solve w = 900.

Word Problems - Answers eLynne wants to save $900 to go to Puerto Rico. She saves $45 each week and now has $180. To find how many more weeks w it will take to have $900, solve w = 900. w=16

7-2 Solving Multi-Step Equations eCombine like terms to simplify an equation before you solve it. eThen solve -- undo addition or subtraction. Then multiply or divide.

Combining Like Terms eM + 2M - 4 = 14 e 3M - 4 = M = M = 6 Example: 7 – y + 5y = 9

Finding Consecutive Integers eConsecutive integers = when you count by 1’s from any integer (ex. 120, 121, 122, 123) eExample: The sum of 3 consecutive integers is 96 eN + (N+1) + (N+2) = 96

Using the Distributive Property e2(5x - 3) = 14 e38 = (-3)(4y + 2) + y e-3(m - 6) = 4 e3(x + 12) - x = 8

Using the Distributive Property - Answers e2(5x - 3) = 14 X=2 e38 = (-3)(4y + 2) + y y= -4 e-3(m - 6) = 4 m= 4 2/3 e3(x + 12) - x = 8 x= -14

7-3 Multi-Step Equations with Fractions and Decimals When there is a fraction next to a variable, you can do the reciprocal to solve the equation

Examples 2 n - 6 = (7/10)k + 14 = (-21) 2/3(m - 6) = 3

Examples - Answers 2 n -6 = 22 3 n= 42 -(7/10)k + 14 = (-21) k=50 2/3(m - 6) = 3 m= 10 1/2

Word Problems Suppose your cell phone plan has $20 per month plus $0.15 per minute. Your bill is $ Use the equation x = How many minutes are on your bill?

Word Problems - Answers Suppose your cell phone plan has $20 per month plus $0.15 per minute. Your bill is $ Use the equation x = How many minutes are on your bill? x=115

7-4 Write an Equation eFive times a number decreases by 11 is 9. eFind the number such that three times the number increased by 7 is 52. eFind a number such that seven less than twice the number is 43.

7-4 Write an Equation - Answers eFive times a number decreases by 11 is 9. 5n - 11 = 19 n = 6 eFind the number such that three times the number increased by 7 is 52. e3n + 7 = 52 n = 15 eFind a number such that seven less than twice the number is 43. e2n - 7 = 43 n = 25

eFifteen more than the product of 8 and a number is -17. eNegative three times a number less four is 17. eThe product of 5 and a number increased by 10 is 145.

eFifteen more than the product of 8 and a number is -17. e15 + 8n = -17 n = -4 eNegative three times a number less four is n - 4 = 17 n = -7 eThe product of 5 and a number increased by 10 is 145. e5n + 10 = 145 n= 27 Answers

eThe difference between half a number and 9 is -23. dThe quotient of a number and 5, diminished by 11 is 18.

Answers eThe difference between half a number and 9 is -23. e1/2n - 9 = -23 n = -28 dThe quotient of a number and 5, diminished by 11 is 18. dN/ = 18 n = 145

eRachel hung 38 ornaments on the tree. This is 3 less than half what Jane hung on the tree. How many ornaments did Jane hang on the tree? eSue did three more than twice the amount of sit-ups that Lisa did. If Sue did 67 sit-ups, how many did Lisa do?

Answers eRachel hung 38 ornaments on the tree. This is 3 less than half what Jane hung on the tree. How many ornaments did Jane hang on the tree? e1/2n - 3 = 38 n = 82 eSue did three more than twice the amount of sit-ups that Lisa did. If Sue did 67 sit-ups, how many did Lisa do? e3 + 2n = 67 n = 32

eFour friends go to dinner together. The check totals $36. They have a coupon for $4 off the total bill. They decide to split the check equally. How much does each person pay?

Answers eFour friends go to dinner together. The check totals $36. They have a coupon for $4 off the total bill. They decide to split the check equally. How much does each person pay? 4x +4 = 36 x = $8 each

eMrs. Mathews has 3 more than twice the number of Christmas pins that Ms. Holden has. If Mrs. Mathews has 39 pins, how many does Ms. Holden have? eThe price of regular set of golf clubs was $ The sale price was $ How much do you save?

Answers eMrs. Mathews has 3 more than twice the number of Christmas pins that Ms. Holden has. If Mrs. Mathews has 39 pins, how many does Ms. Holden have? 2x +3 = 39 x = 18 pins eThe price of regular set of golf clubs was $ The sale price was $ How much do you save? x = x = $66.70

Word Problems Two-thirds the number of girls plus two represents the number of boys in the class. If there are 13 boys in the class, how many girls are there?

Word Problems - Answers Two-thirds the number of girls plus two represents the number of boys in the class. If there are 13 boys in the class, how many girls are there? 2/3y + 2 = 13 y = 16.5

7-5 Solving Equations with Variables on Both Sides To solve an equation with a variable on both sides, use addition or subtraction to collect the variable on one side of the equation.

Collecting the variable on one side 9a + 2 = 4a x + 4 = 2x + 36 k + 9 = 6(k - 11)

Collecting the variable on one side - Answers 9a + 2 = 4a - 18 a = -4 4x + 4 = 2x + 36 x = 16 k + 9 = 6(k - 11) k = 15

Word Problem eBeth leaves home on her bicycle, riding at a steady rate of 8 mi/h. Her brother, Ted, leaves home on his bicycle 1/2 an hour later, following Beth’s route. He rides at a steady rate of 12 mi/h. How long after Beth leaves home will Ted catch up?

Word Problem - Answer eBeth leaves home on her bicycle, riding at a steady rate of 8 mi/h. Her brother, Ted, leaves home on his bicycle 1/2 an hour later, following Beth’s route. He rides at a steady rate of 12 mi/h. How long after Beth leaves home will Ted catch up? 8x = 12(x - 1/2) x = 1.5

7-5 Solving Equations with Variables on both sides (Day 2) 5(w + 3) = 4(w - 2) 9 - d = d

7-5 Solving Equations with Variables on both sides (Day 2) - Answers 5(w + 3) = 4(w - 2) w= d = d d= -11

Word Problems Five more than three times a number is the same as four less than twice a number. Find the number. Sixty-seven, decreased by four times a number, is the same as eight times a number, increased by seven. Find the number.

Word Problems - Answers Five more than three times a number is the same as four less than twice a number. Find the number y = 2y – 4; y= -9 Sixty-seven, decreased by four times a number, is the same as eight times a number, increased by seven. Find the number. 67 – 4a = 8a + 7; a = 5

Consecutive Integers When you count by 1’s from any integer, you are counting consecutive integers Example: 45, 46, 47 When you count by 2’s from any number you are counting either consecutive odd or even integers Example: 2, 4, 6 or 3, 5, 7

Finding Consecutive Integers The sum of 3 consecutive integers is 96. Find the numbers. Find two consecutive even integers with a sum of 66. Find 2 consecutive even integers such that the sum of the larger and twice the smaller is 38.

Finding Consecutive Integers - Answers The sum of 3 consecutive integers is 96. Find the numbers. n + (n+1) + (n+2)=96; 31, 32, 33 Find two consecutive even integers with a sum of 66. n + (n+2) = 66; 32,34 Find 2 consecutive even integers such that the sum of the larger and twice the smaller is 38. 2n + (n +2) = 38; 12, 14

Find the value of x and the perimeter The square and the triangle have equal perimeters. A.Find the value of x B.Find the perimeter (Square: Side is x-3) (Triangle: Sides are x, x, and 8)

Find the value of x and the perimeter The square and the triangle have equal perimeters. A.Find the value of x 4(x - 3) = x + x + 8; x = 10 B. Find the perimeter p = 28 (Square: Side is x-3) (Triangle: Sides are x, x, and 8)

Find the missing value The Yellow Bus Company charges $160 plus $80 per hour to rent a bus. The Orange Bus Company charges $200 plus $60 per hour. A.For what number of hours would the companies charge the same? B.What would the charge be for that number of hours?

Find the missing value The Yellow Bus Company charges $160 plus $80 per hour to rent a bus. The Orange Bus Company charges $200 plus $60 per hour. A.For what number of hours would the companies charge the same? h = h; h = 2 hours B.What would the charge be for that number of hours? $320

7-7 Transforming Formulas You can use the properties of equality to transform a formula to represent one quantity in terms of another.

Transforming in one step Solve the area formula A = lw for l Examples: p = s - c (solve for s) h = k/j (solve for k)

Transforming in one step - Answers Solve the area formula A = lw for l l = A/w Examples: p = s - c s = p + c h = k/j k = h x j

Using more than one step Solve the formula P = 2L + 2W for L Y = 3/5p - 4 solve for p R = n(C - F) solve for C

Using more than one step - Answers Solve the formula P = 2L + 2W for L l = (P-2w)/2 Y = 3/5p - 4 solve for p p = 5/3(y + 4) R = n(C - F) solve for C C = (R + nF)/n

7-8 Simple and Compound Interest When you first deposit money in a savings account, your deposit is called principal. The bank takes the money and invests it. In return, the bank pays you interest based on interest rates. Simple interest is interest paid only on the principal.

Simple interest formula I = prt I = interest P = principal R = interest rate per year T = time in years

Compound interest When a bank pays interest on the principal and on the interest an account has earned, the bank is paying compound interest. The principal plus the interest is the balance, which becomes the principal on which the bank figures the next interest payment.

Compound Interest Formula Formula --- B = p(1 + r) n B = final balance P = principal R = interest rate for each interest period N = number of interest periods

Interest and Interest Periods Semiannually = 2 times a year Semiannually for 2 years = 4 interest period Semiannually at 3% = 3/2 = 1.5