Chapter 3 Solving Linear Equations

1. Chapter 3 Solving Linear Equations 3.1 Solving Equations using addition and subtraction Ex: 8 – y = -9 8 – y – 8 = - 9 – 8 to isolate the variable subtract 8 from both sides -y = -17 (-1) (-y) = (-17) (-1) multiply both sides by (-1) y = 17 8 – 17 = -9 answer check Ex: |-14| +z = 12

2. 3. 2 Solving Equations by multiplication and division Ex: n = -3 -8 8 Ex: ¼ (y + 8) = 5

3. 3.3 Solving Multi-Step Equations Isolate the variable Ex: (1/2) x- 5 = 10 Ex: 10(2-x) + 4x = (-3/10)(x+3)

4. Ex: In any triangle, the sum of the measure of the angles is 180 degrees. In triangle ABC, <A is 3 times as large as B. Angle <C measures 20 degrees less than <B. Find <A, <B, and <C. What do we know: <A + <B + <C =180 <A=3 <B <C=<B-20 (3<B) +<B + (<B-20) = 180 Substitute 5<B -20 = 180 Combine like terms +20 +20 5<B =200 5<B =200 5 5 <B=40 If <B =40, then <A=3<B or <A= 3(40). <A=120 If <B =40, then <C=<B-20 or <C=40-20. <C=20 To check, <A +<B +<C must =180; 120 +40 +20 =180 check

5. Coin problems: Ex There are 4 times as many nickels as dimes in a coin bank. The coins have a total value of 600 cents ($6.00). Find the number of nickels. What do we know: let n=nickels d=dimes 5n+10d=600 n=4d 5(4d) +10d =600 20d + 10d = 600 30d = 600 d=20 but the question asks for the # of nickels so n=4d d=20 n=4(20)= 80 nickels 5(80) +10(20) = 600 yes

6. 3.3 p 149 #51 Number problem The sum of three numbers is 123. The second number is 9 less than 2 times the first number. The third number is 6 more than 3 times the first number. Find all three numbers. What do we know: x + y + z = 123 y=2x-9 z=3x+6 x + (2x-9) + (3x+6) = 123 6x -3 = 123 6x = 126 x=21 If x = 21, then y=2(21)-9. y=33 If x=21, then z=3(21)+6. z=69 To check: 21+33+69=123 yes Answer: 21, 33, 69

7. Quiz Thursday, 3.1-3.3, know the problems from today. By Thursday you should have completed or at least attempted: 3.1, 3.2, 3.3, 3.3 2nd day What is due on Friday: 3.1, 3.2, 3.3, 3.3 2nd day, and 3.4 1st day

8. 3.4 Solving Equations with Variables on Both Sides Collect on the left side: 6x+22=-3x+31 Collect on the right side: 64-10w=6w Identity (always true): 4(x-5)=4x-20 No solution (never true) 3x-9=3x+10

9. 3.4 more complicated examples 10(2-x)+4x= -(3/10)(x+3) (2/5)(10x+15)=18-4(x-3)

10. 3.4 word problem A gym offers two packages for yearly membership. The first plan costs $50 to be a member. Then each visit to the gym is $5. The second plan costs $200 for a membership fee plus $2 per visit. Which membership is more economical? 50 + 5x =200 + 2x -2x -2x 50 + 3x = 200 -50 -50 3x = 150 x=50 Plan #1 is better for less than 50 visits. Plan #2 is better for more than 50 visits

11. 3.5 Linear Equations and Problem Solving Steps to solving a word problem: 1. Understand the problem-summarize in own words 2. Visualize the problem (picture, diagram, table) 3. Verbal Model 4. Label knowns and unknown 5. Algebraic model 6. Solve 7. Check solution

12. A gazelle can run 73 feet per second for several minutes. A cheetah can run faster (88 ft per second) but can only sustain its top speed for about 20 seconds before it is worn out. How far away from the cheetah does the gazelle need to stay for it to be safe? Gazelle distance in 20 secs + Gazelle’s starting distance = Distance Cheetah can run in 20 seconds 73(20) + x = 88(20) 1460 + x = 1760 x = 300 feet Gazelle and cheetah

13. Hmwk: 3.5 p164 #6-13,17,22,35,36,40

14. Homework review 3.5 #17 Current Japanese students + additional Japanese students= Current German students – loss of students 45 +3x = 108 -4x +4x+4x 45 +7x= 108 -45-45 7x=63 x=9 years

15. 3.6 Linear Equations and Problem Solving Ex 1: If four people are sharing the cost of a monthly phone bill of $58.25, what is each person’s share of the bill? 4x=58.25 x=58.25 4 x=14.5625 x=$14.56 per person

16. Ex 2: Solve 26x-32 = 99 +32+32 26x = 131 x = 131/26 x =5.03846… x=5.04 Ex 3: Solve 9.92x-6.13 = 5.96 – 7.28x 992x-613 = 596 – 728x (multiply by 100) +613+613 992x=1209 – 728x +728x+728x 1720x=1209 x=.70290… rounds to .70

17. Ex 4: Solve 3.7x – 2.5 = 6.1x-12.1 37x – 25 = 61x – 121 (multiply by 10) +121+121 37x -96 = 61x -37x-37x 96 = 24x 4=x

18. Ex 5: You are shopping for a halloween costume. Sales tax is 6%. You have $21.25 to spend. What is the price limit of the costume? Costume + costume (sales tax) = $21.25 x + .06x = $21.25 1.06 x = 21.25 106 x = 2125 x=20.0471698… x=$20.04 for the costume (why not $20.05) Hmwk: 3.6 p169 #14,16,20,24,28,32,36,40,42,53

19. 3.7 Formulas and Functions Formula= an algebraic equation that relates two or more real life quantities • Solve A=LW for W L L A/L=W 2. Solve K=(5/9)(F-32) +273 for F (9/5)K= (9/5)(5/9)(F-32)+(9/5)(273) -(9/5)(273)-(9/5)(273) (9/5)K-(9/5)(273) =F-32 (9/5)(K-273) +32 = F

20. Ex 3: Solve I=prt for t t=I/pr Find the number of years (t) that $2800 was invested to earn $504 at 4.5% t=I/pr t=504/(2800)(.045) t=4 years

21. Rewriting equations in function form A two variable equation is in function form if the output variable is isolated on one side Ex: Rewrite the equation 2x-y=9 so that y is a function of x. 2x-y=9 Write x as a function of y -2x-2x -y=9-2x (-1)(-y)= (-1)(9-2x) y= -9+2x y=2x-9

22. Rewrite y=2x-9 as a function of x +9+9 y+9 = 2x (y+9) = x 2 Use the results to find x when y=-2,-1,0,1

23. Hmwk: 3.7 p177 #11-14, 16-20,24,28,31,37,38,44

24. 3.7 Homework examples p 177 # 11) Area of a triangle, solve for B: A=(1/2)bh 2A=(1/2)bh(2) 2A=bh 2A=bh h h (2A/h)=b

25. p 177 #14 Area of a trapezoid: solve for b2 A=(1/2)(h)(b1+b2) 2A=(2)(1/2)(h)(b1+b2) 2A=(h)(b1+b2) 2A= hb1+ hb2 2A-hb1=hb2 2A-hb1= hb2 h h (2A/h) –b1= b2