Transcript Solving Linear Equations

Finding the Unknown



Equation: A mathematical sentence with an
equals sign, which states that 2 expressions
are equal:
12 – 3 = 9
18 + 6 = 24
Balance these equations so that both sides are equal
Equals
Left Hand Side
Right Hand Side
12+ 4 + 6
9+7+2
3 – 13 + 5
= 21
=6+2
=3+2

An Equation is like a balance scale.
Everything must be equal on both sides
If we change one side of
the equation or balance
(by adding or subtracting)
we must also do the same
to the other side.
 If we don't the scales
become unbalanced and
are no longer equal!

Find the value of x for each of the following
10x + 20
5x + 5
6x + 30
2x + 12
= 40
= 20
= 36
= 20
How can we solve for X easier?
Use Inverse Operations!
Inverse Operation : An operation that reverses the effect of
another operation. Some simple inverse operation are
below.
Operation
Addition
Inverse Operation
Subtraction
Division
Multiplication
One Step Equations
To solve one step equations, you need to
ask three questions about the equation:
1
What is the variable?
2
What operation is performed on the variable?
3
What is the inverse operation?
(The one that will undo what is being done to
the variable)
Example 1 Solve x + 4 = 12
What is the variable? The variable is x.
What operation is being performed on the variable?
Addition.
What is the inverse operation (the one that will undo what is being
done to the variable)?
Subtraction.
Using the subtraction property of equality, subtract 4 from both
sides of the equation.
The subtraction property of equality tells us to subtract the same thing on
both sides to keep the equation equal.
x + 4 = 12
-4
-4
x = 8
4a = 24
This means
“4 multiplied by a”
What is the inverse
operation?
Division
a = 24
4
=6
This means
“b divided by 5”
What is the
inverse
operation?
Multiplication
b=3
5
b=3x5
= 15
Two Step Equations
The Problem:
This problem
has addition, so
we need to
subtract first.
8x + 5 = 61
-5 -5
= 56
8x
Remember: Whatever we do on one side, we have to do on the
other.
This problem has
multiplication, so
we need to
divide now.
8x = 56
8
8
x = 7
Remember: Whatever we do on one side, we have to do on the
other.
This problem has
subtraction, so
we need to add
first.
3a - 8 = 4
+8 +8
3a
= 12
Remember: Whatever we do on one side, we have to do on the
other.
This problem has
Multiplication, so
we need to
divide now .
3a = 12
3
3
a = 4
Remember: Whatever we do on one side, we have to do on the
other.
Let’s make this
question into a simpler
This is the onlyThis
bit says
that
expression. We know
is different. We
can8”
use
“add
how to solve those!
inverse operations to
What
get ridisofthe
it. inverse
operation?
Subtraction
Look! Now your
question looks
simple to answer!
3a + 8 = 35
3a = 35 - 8
3a = 27
a = 27
3
=9

How to solve worded problems:
1. Identify the unknown quantity and use a
2.
3.
4.
5.
Pronumeral to represent it.
Search for keywords that indicate the steps
needed for the solution.
Create a linear equation from the information
provided in the question.
Solve the equation.
Interpret the result and write the worded
answer.
Word Problem
Linear Equation
What is the sum of 8 and y? 8 + y
Express the number (x) of
x+2
apples increased by two
Express the total weight of
Alphie the dog (x) x + y
and Cyrus the cat (y)
Key words for addition + :
increased by; more than; combined together; total of; sum;
added to
Word Problem
What is four less than y
What is nine less than a
number (y)
What if the number (x) of
children was reduced by 6?
What is the difference of my
weight (x)
and your weight (y)
Linear Equation
y-4
y-9
x-6
x-y
Key words for Subtraction - :
less than, fewer than, reduced by, decreased by, difference of
Word Problem
Linear Equation
What is y multiplied by 13 13y or 13 * y
Three runners averaged "y"
minutes.
3y
Express their total running
time:
I drive my car at 55 miles per
hour. 55x
How far will I go in "x" hours?
Key words for multiplication * x
or integers next to each other (5y, xy) :
of, times, multiplied by
Word Problem
Linear Equation
What is the quotient of y and
y/3 or y ÷ 3
3
Three students rent an
apartment for $ "x" /month. x/3 or x ÷ 3
What will each have to pay?
"y" items cost a total
of $25.00. 25/y or 25 ÷ y
Express their average cost:
Key words for division ÷ /
per, a; out of; ratio of, quotient of; percent (divide by 100)
Al's father is 45. He is 15 years older than twice Al's age.
How old is Al?
 We can begin by assigning a Variable/Pronumeral to what
we're asked to find. Here this is Al's age, so let Al's age = x.
 We also know from the information given in the problem
that 45 is 15 more than twice Al's age.

 How can we translate this from words into mathematical
symbols?
 What is twice Al's age?
Well, Al's age is x, so twice Al's age is 2x, and 15 more
than twice Al's age is 15 + 2x. That equals 45, right?
 Now we have an equation in terms of one variable that we
can solve for x:
 45 = 15 + 2x.

Original Problem: 45 = 15 + 2x
First step is to get rid of the number 15. To do this we need to subtract 15
from both sides of the equation.
45 – 15 = 15 – 15 + 2x
We now have: 30 = 2x
The Second Step is to get the variable/Pronumeral by itself (in this case a
single x). To do this we will need to divide 2x by 2, which means we have
to divide both sides of the equation by 2.
15 = x
or
x = 15
Since x is Al's age and x = 15, this means that Al is 15 years old.
Practice these word problems on your own:
There are 26 students in Ms. Bean's class. The number
of boys is equal to seven fewer than twice the
number of girls. How many boys and how many girls
are in the class?
You are ordering tulip bulbs from a flowering catalog.
The cost is .75 cents per bulb. You have $14 to
spend. If the shipping cost is $3 for any size order,
determine the number of bulbs you can order.
x + 2x – 7 = 26 x is # of girls; 2x-7 is # of boys
3x – 7 = 26
+7 +7 Addition Property of Equality
3x = 33 Division Property of Equality
3 3
x = 11
So there are 11 girls and 15 boys.
.75b + 3 = 14
- 3 - 3 Addition Property of Equality
.75b = 11 Division Property of Equality
.75 .75
b = 14.667
So you can purchase about 14 bulbs. If you
purchase 15 bulbs you will go over your $14
budget.