Transcript UnitII Equations

Unit II
Equations
Solving Equations
What is an equation?
A mathematical statement that two expressions are equal
When solving any equation you want to ISOLATE THE VARIABLE.
A variable is isolated when it appears by itself on one side of an equation
and not all on the other side. Isolate a variable by using inverse operations, which
“undo” operations on the variable.
An equation is like a balanced scale. To keep the balance, you must
perform the same operation on both sides.
Inverse Operations
Add x
Subtract x
Multiply x
Divide x
Solving Equations
Algebraic equations are mathematical statements that two expressions are
equal to each other.
x 5 8
When solving for a variable, you must ISOLATE the variable, meaning
get the variable alone.
• Check to see if you can simplify either side. (combine like terms,
distribute etc.)
• Get all of the variables to one side. (Hint: Try moving the “smaller”
variable term. )
• Move all of the numbers to the other side by performing the
opposite operations.
• CHECK your solution to make sure it works. (Replace the variable
with the number you got as the solution.)
Solving Equations using Addition & Subtraction
x 10  4
+ 10
Since 10 is subtracted from x, add 10 to both sides to undo the subtraction.
+10
x  14
x7 9
-7
Since 7 is added to x, subtract 7 from both sides to undo the addition.
-7
x  2
Try these:
x 15  32
47
x  21  14
7
x 5  0
x 17  24
5
7
Solving Equations using Multiplication & Division
(-5)
x
4
5
(-5)
Since x is divided by -5, multiply both sides by -5 to undo the division.
x  20
12
x
48
____
____
12
Since x is multiplied by 12, divide both sides by 12 to undo the multiplication.
12
x  4
Try these:
x
2
14
9x  81
28
9
y
 12
5
60
100  20 y
5
Solving Multi-Step Equations
10  6  2x
-6
-6
4  ____
2x
____
-2
-2
Do any adding or subtracting first! Remember, you want to ISOLATE the
variable.
Multiplication or Division comes next!
2  x
6x  3  8x  13
2x  3  13
Simplify each side of the equation if possible, combine like terms.
2x  10
Add or subtract, remembering to move constants AWAY from the
Variable.
x  5
Multiply or Divide to get the variable alone.
Try these:
1) 2(7  c)  6
x  10
x 3
2) 4x  6x  30
3)
z
3
1 
2
2
z 1
7)
3
a  14  8
4
a  8
8) 7 x  2  5x  8
x 3
9) 4(2 x  5)  5 x  4
x 8
1
2
4) 17  3( p  5)  8
p 8
10) 6  7(a  1)  3(2  a)
a
5) x  (12  x)  38
x  25
11) 4(3x  1)  7 x  6  5 x  2
all real #s
6) z  5  7
6 8
8
z 9
12)
2
(3x  9)  8 x
3
x 1
When solving an EQUATION:
GOAL: ISOLATE the variable (get the variable alone).
• Check to see if you can simplify either side. (distribute, combine
like terms etc.)
• Get all of the variables to one side (using inverse operations).
• Move all of the numbers to the other side (using inverse operations)
• CHECK your solution to make sure it works. (Replace the variable
with the number you get as the solution.)
Applications
1) Stephen belongs to a movie club in which he pays an annual fee of $39.95 and
then rents DVDs for $0.99 each. In one year, Stephen spent $55.79. Write and
solve an equation to find how many DVDs he rented.
Equation:
39.95 + 0.99 d= 55.79
Answer:
16 DVDs
2) Maggie’s brother is three years younger than twice her age. The sum of their ages
is 24. How old is Maggie?
Answer:
9 years old
3) The sum of two consecutive whole numbers is 57. What are the two numbers?
Answer: 28 and 29
4) The height of an ostrich is 20 inches more than 4 times the height of a kiwi. Write
and solve an equation to find the height of a kiwi.
Equation: 4k+20 = 108
Answer: 22 inches
Solving Absolute Value Equations
Remember that the absolute value of a number is that number’s distance from zero on a
number line.
5  5 and 5  5
For any nonzero absolute value, there are exactly two numbers with that absolute value.
For examples, both 5 and -5 have an absolute value of 5.
To write this statement using algebra, you would write x  5 . This equation asks, “What
values of x have an absolute value of 5?” The solutions are 5 and -5.
Ex. A
4 x  2  24
_____ ___
4
4
x2 6
The solutions are 4 and -8.
Since x  2 is multiplied by 4, divide both sides by 4 to undo the multiplication.
Rewrite the equation as two cases. Since 2 is added to x, subtract 2 from both
sides of the equation.
Case 1 x  2  6
-2 -2
x4
Case 2
x  2  6
-2
x  8
* Remember, absolute value cannot be negative!
-2
Solving Proportions
Proportion: a statement that two ratios are equal
* Cross multiply to solve
Ex. A
5 y

2 8
(5)(8)  (2)( y )
40  2 y
20  y
Ex. B
x3 7

12
2
2( x  3)  (12)(7)
2x  6  84
2x  78
x  39
a c

b d
Absolute Value & Proportion Practice
1) 9  x  5
2)
3x  5 x

14
3
3) 2 x  3  18
4)
5t  3 t  3

2
2
5)
2 x  4  22
14, 4
6) 5 x  7  14  8
no solution
x 3
7)
x 1 x 1

3
5
6, 12
8)
3x  9  7  7
3
0
9)
5
8

2n 3n  24
120
10)
2x  5.75  13.25
9,13
x4
9.5,3.75
Percents
A percent is a ratio that compared a number to 100. For example, 25% =
Fraction equivalent of a %:
Finding the Part
Ex. A
Find 50% of 20.
part
percent

whole
100
14
x

24 100
24x  1400
x  58.3
Decimal equivalent of a %:
25% 
25% 
Use the percent proportion.
Let x represent the percent.
Find the cross products.
Since x is multiplied by 24, divide both
sides by 24 to undo the multiplication.
25
100
25 1

100 4
25
 0.25
100
Percent Practice
1) What % of 60 is 15?
25
2) 440 is what % of 400?
7) Kate found a new dress at the mall. The
price tag reads $90. The sign above the rack
of dresses says that all items are 40% off.
How much will Kate pay for the dress?
110%
$54
3) 40% of what number is 14?
4) Find 105% of 72.
35
8) On average, sloths spend 16.5 hours per
day sleeping. What percent of the day do
sloths spend sleeping? Round your answer
to the nearest percent.
69%
75.6
5) 36 is 90% of what number?
6) 5 is what percent of 50?
40
10
9) A can of ice tea contains 4% of the
recommended daily allowance of sodium.
The recommended daily allowance is 2500 mg
How many milligrams are in the can of ice tea?
100 mg
10) A newspaper reported that 42% of
Registered voters, voted in the election. If
12,000 people voted, how many registered
voters are there?
28,572
1) A taxi company charges $2.10 plus $0.80 per mile. Carmen paid a fare of $11.70. Write and
solve an equation to find the number of miles she traveled.
$2.10 + .80m=$11.70
m = 12 miles
2) On the first day of the year, David had $700 in his savings account and started spending $35
a week. His brother John had $450 and started saving $15 a week. After how many weeks
will the brothers have the same amount? What will that amount be?
5 weeks
$525
3) Peter earns $32,000 per year plus a 2.5% commission on his jewelry sales. Find Peter’s total
salary for the year when his sales are valued at $420,000.
$42,500
4) A volunteer at the zoo is responsible for feeding the animals in 15 exhibits in the reptile house.
This represents 20% of the total exhibits in the reptile house. How many exhibits are in the
reptile house?
75 exhibits