Download File - mathemons

Solving
Linear
Equations
Bellwork
Solve for x:
1.
x – 5 = 57
2.
3 – x = 10
3. 4x + 28 = 68
4. -3x – 7 = 14
5. 4x – 9 + x = 16
Problem No. 1
1.
Mica bought a CD for P25
and 8 blank videotapes. The
total cost was P265. Find the
cost of each blank videotape.
Equation: 25 + 8x = 265
Solution:
25 + 8x = 265
8x = 265 – 25
8x = 240
8
8
Answer: P30 each videotape
Problem No. 2
2. Juan bought 5 orchids
in pots and P200 rose
plant at a fund raiser. He
spent a total of P750. Find
the cost of each orchid.
Equation: 5x + 200 = 750
Solution:
5x + 200 = 750
5x = 750 – 200
5x = 550
5
5
x = 110
Answer: The price of each orchid is P110.
Error Analysis
10x + 4 = -2
10x = 2
10x = 2
10 10
x = 2 or 1
10
5
Correct Solution:
10x + 4 = -2
10x = -2 – 4
10x = -6
10x = -6
10 10
x = -3/5
Topic: Solving One Step Equations
Addition Property of Equality – A property
that states that we must add the same number
on both sides of the equation to make it equal
or balance.
Example:
x–4=7
x–4+4=7+4
x = 11
Subtraction Property of Equality
– A property that states that we
must subtract the same number
on both sides of the equation to
make it equal or balance.
x+2=8
x+2–2=8–2
x=6
Multiplication Property of Equality
– A property that states that we
must multiply the same number on
both sides of the equation to make it
equal or balance.
½x=8
2 (1/2 x) = 8 (2)
x = 16
Division Property of Equality – A
property that states that we must
divide the same number on both
sides of the equation to make it
equal or balance.
6x = 36
6x = 36
6
6
x=6
Topic: Solving Two-Step Equations
In order to solve two-step linear equations
you need to use properties of equality.
Example No. 1
2x – 4 = 10
2x = 10 + 4
2x = 14
2
2
x=7
Given
Isolate term with x
Divide both sides by 2
Example No. 2
3x + 8 = 14
Given
3x = 14 – 8
Isolate the term with x
3x = 6
Divide both sides by 3
3 3
x=2
Checking: Substitute x = 2 in the equation
3x + 8 = 14
3(2) + 8 = 14
6 + 8 = 14 Therefore it is correct!
Topic: Solving Multi-Step Equations
In order to solve multi-step equations you need
to use your knowledge about distributive
property and combining like terms.
Example No. 1
2x + 3x – 5 = 35
Given
5x – 5 = 35
Add like terms
5x = 35 + 5
Add 5 on both sides
5x = 40
Divide both sides by 5
5 5
x=8
Example No. 2
3 ( x – 4 ) = 24
Given
3x – 12 = 24 Use Distributive Property
3x = 24 + 12 Add 12 on both sides
3x = 36
Divide both sides by 3
x = 12
3 3
Checking: Substitute x = 12 in the given eq.
3 ( x – 4 ) = 24
3 (12 – 4 ) = 24
3 (8) = 24 Therefore it is correct!
Topic: Solving Equations with
Variables on Both Sides.
To solve an equation with variables on both
sides, you need to put all the variable terms on
one side.
Example No. 1
9x + 2 = 4x – 18
Given
9x – 4x = -18 – 2 Isolate variable terms
5x = -20
Divide both sides by 5
5
5
x = -4
Example No. 2
5x – 8 = - 2x + 6
5x + 2x = 6 + 8
7x = 14
7
7
Given
Isolate variable terms
Divide both sides by 7
x=2
Checking: Substitute x = 2 in the given eq.
5x – 8 = -2x + 6
5(2) – 8 = -2(2) + 6
10 – 8 = -4 + 6
2 = 2 Therefore it is correct!
Board Drill
4x – 10 + 6x = 100
Board Drill
7 ( x – 5 ) = 21
Board Drill
3x + x – 2 = 3
Board Drill
-6x + 3x – 9 = 18
Board Drill
9x + 10 = 2x + 31
Board Drill
-4x + 7 = 6x - 3
Board Drill
6x - 9 = x + 36
Board Drill
7x + 9 = 3x + 25
Board Drill
5x + 8 = 7x
Topic: Word Problems Involving
Linear Equations
A verbal problem or a
mathematically-worded problem is
a problem of mathematical nature
stated in plain words, and which
would involve mathematical
calculation of some kind before it
can be solved.
Note:
There are no set of rules or methods
Which enable us to solve all kinds of
Problems, because things must be
Remembered in relation to different
Types of problems.
The following are the general
strategies for problem solving.
1. Read the problem carefully. Be sure
that you understand what the problem is
all about.
2. Take note of what is asked in the
problem.
3. Represent the unknown by any
variable and other unknowns in terms
of the same variable according to the
conditions of the problem.
4. Formulate the equation.
5. Solve the resulting equation.
6. Check your answer by substituting
it to the original equation and check if
your answer or answers satisfy the
problem.
Examples
Number Relation Problems
The sum of two numbers is
36. One number is 3 less than
twice the other number. What
are the numbers
Representation:
let x = the other number
2x – 3 = one number
Equation:
x + 2x – 3 = 36
Solution:
3x – 3 = 36
3x = 36 + 3
3x = 39
Answers: 13 and 23
Another Example
Seven more than twice a
number is four less than
thrice the number. What
is the number?
Example
Consecutive Numbers Problems
The sum of three
consecutive numbers is
135. Find the numbers.
Representation:
let x = 1st number
x + 1 = 2nd number
x + 2 = 3rd number
Equation:
x + x + 1 + x + 2 = 135
Solution:
3x = 135 – 3
3x = 132
Answers: 44, 45, and 46
Another Example
Find three
consecutive
numbers whose
sum is 60.
Example
Age Problems
Lherry is 3 times as old as
Jane. In 4 years time, Lherry
will be twice as old as Jane.
How old is Jane?
Representation:
Present Age
x
3x
Jane
Lherry
Equation:
3x + 4 = 2 ( x + 4)
Age in 4 years
x+4
3x + 4
Solution:
3x + 4 = 2 (x + 4)
3x + 4 = 2x + 8
3x – 2x = 8 – 4
x=4
Answer:
Jane is 4 years old while Lherry is
12 years old.
Another Example
Paulson is four times
older than Maria. If the
total of their ages is 60.
How old is Maria?
Board Drill
Thirty more than
thrice a number is
45. What is the
number?
Board Drill
Ten more than twice
a number is 100.
What is the
number?
Board Drill
The sum of three
consecutive
numbers is 93. Find
the numbers.
Board Drill
Daisy, 38 years old, is
8 years more than
three times as old as
her son. How old is
her son?
Board Drill
Leah is three times
older than Marvin. If
the total of their ages
is 68. How old is
Marvin?
Board Drill
A moving van rents for P1,250
a day plus P5 per kilometer.
Mrs. Santos’ bill for a two day
rental was P2000. How many
kilometers did she drive?