Solving Linear Equations

Finding the Unknown

Definitions

Equation: A mathematical sentence with an equals sign, which states that 2 expressions are equal:

12 – 3 = 9 18 + 6 = 24

Balance these equations

Left Hand Side Right Hand Side12+ 4 + 6 = 219 + 7 + 2 = 6 + 2

3 – 13 + 5 = 3 + 2

Balance these equations so that both sides are equal Equals

Activity: Balance the Scales

Scales in Balance

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!

Finding an Unknown

10x + 20 = 40 5x + 5 = 20

6x + 30 = 362x + 12 = 20

Find the value of x for each of the following

How can we solve for X easier?

Use Inverse Operations!

Inverse Operations

Operation Inverse Operation

Addition Subtraction

Division Multiplication

Inverse Operation : An operation that reverses the effect of another operation. Some simple inverse operation are below.

Solving equations with an unknown on one side

One Step Equations

ONE STEP EQUATIONS

What is the variable?

To solve one step equations, you need to ask three questions about the equation:

What operation is performed on the variable?

What is the inverse operation?

(The one that will undo what is being done to the variable)

1

2

3

ONE STEP EQUATIONS

Example 1 Solve x + 4 = 12What is the variable?

Using the subtraction property of equality, subtract 4 from both sides of the equation.

- 4

x

x + 4 = 12

The variable is x.Addition.

Subtraction.What is the inverse operation (the one that will undo what is being done to the variable)?

What operation is being performed on the variable?

- 48=

The subtraction property of equality tells us to subtract the same thing on both sides to keep the equation equal.

Example 1

4a = 24This means

“4 multiplied by a”

What is the inverse operation?

Division

a = 24 4

= 6

Example 2

b = 35

This means

“b divided by 5”

What is the inverse

operation?

Multiplication

b = 3 x 5

= 15

Solving equations with an unknown on one side

Two Step Equations

Let’s try a problem.

8x + 5 = 61

This problem has addition, so we need to subtract first.

- 5 - 5

Remember: Whatever we do on one side, we have to do on the other.

8x

56=

The Problem:

Now, Step 2.

8x = 568 8

x = 7Remember: Whatever we do on one side, we have to do on the other.

This problem has multiplication, so we need to divide now.

Let’s try another problem.

3a - 8 = 4+ 8 + 8

3a

12=

This problem has subtraction, so we need to add

first.

Remember: Whatever we do on one side, we have to do on the other.

Now, Step 2.

3a = 123 3

a = 4This problem has Multiplication, so

we need to divide now .

Remember: Whatever we do on one side, we have to do on the other.

Example 3

3a + 8 = 35This is the only bit that

is different. We can use inverse operations to get rid of it.

This says “add 8”

What is the inverse operation?

Subtraction

3a = 35 - 83a = 27

Look! Now your question looks simple to answer!

a = 27 3

= 9

Let’s make this question into a simpler expression. We know how to solve those!

Solving word problems using linear equations

Solving Word Problems

How to solve worded problems:1. Identify the unknown quantity and use a

Pronumeral to represent it.2. Search for keywords that indicate the

steps needed for the solution.3. Create a linear equation from the

information provided in the question.4. Solve the equation.5. Interpret the result and write the worded

answer.

Addition

Word Problem Linear Equation What is the sum of 8

and y?8 + y

Express the number (x) of apples

increased by twox + 2

Express the total weight of Alphie the

dog (x) and Cyrus the cat (y)

x + yKey words for addition + :increased by; more than; combined together; total of; sum; added to

Subtraction

Key words for Subtraction - :less than, fewer than, reduced by, decreased by, difference of

Word Problem Linear Equation What is four less than

yy - 4

What is nine less than a number (y)

y - 9

What if the number (x) of children was

reduced by 6?x - 6

What is the difference of my weight (x)

and your weight (y)x - y

Multiplication

Key words for multiplication * x or integers next to each other (5y, xy) :of, times, multiplied by

Word Problem Linear Equation What is y multiplied

by 1313y or 13 * y

Three runners averaged "y" minutes.

Express their total running time:

3y

I drive my car at 55 miles per hour.

How far will I go in "x" hours?

55x

Division

Key words for division ÷ /per, a; out of; ratio of, quotient of; percent (divide by 100)

Word Problem Linear Equation What is the quotient

of y and 3y/3 or y ÷ 3

Three students rent an apartment for $ "x"

/month.What will each have to

pay?

x/3 or x ÷ 3

"y" items cost a total of  $25.00.

Express their average cost:

25/y or 25 ÷ y

Complex question example 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.

The Solution – Finding the value of x

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.

Real World Situations

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.

Ms Bean’s Class

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 = 11So there are 11 girls and 15 boys.

Ordering Tulips

.75b + 3 = 14 - 3 - 3 Addition Property of Equality

.75b = 11 Division Property of Equality

.75 .75 b = 14.667So you can purchase about 14 bulbs.

If you purchase 15 bulbs you will go over your $14 budget.