Download Chapter 3 Solving Linear Equations

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
3. 2 Solving Equations by multiplication and division
Ex: n = -3
-8 8
Ex: ¼ (y + 8) = 5
3.3 Solving Multi-Step Equations
Isolate the variable
Ex: (1/2) x- 5 = 10
Ex: 10(2-x) + 4x = (-3/10)(x+3)
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
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
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
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
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
3.4 more complicated examples
10(2-x)+4x= -(3/10)(x+3)
(2/5)(10x+15)=18-4(x-3)
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
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
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?
2000
1500
1000
500
0
Time
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
Time
C-88
G-73
0
0
x
5
5(88)
5(73)+x
10
10(88)
10(73)+x
15
15(88)
15(73)+x
20
20(88)
20(73)+x
Hmwk: 3.5 p164 #6-13,17,22,35,36,40
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
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
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
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
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
3.7 Formulas and Functions
Formula= an algebraic equation that relates two or
more real life quantities
1. 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
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
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
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
Y
-2
-1
0
1
(y+9)/2
(-2+9)/2
x
3.5
Hmwk: 3.7 p177 #11-14, 16-20,24,28,31,37,38,44
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
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