Variable. the Constant Is the ______That Is Written by Itself

1

/ MATHEMATICS 8
CHAPTER 10– SOLVING LINEAR EQUATIONS
TRANSLATING ENGLISH INTO ALGEBRA / Date ______

In an algebraic expression, the variable is an ______value and is usually represented by a ______(most often, it is “_____”). The coefficient is the ______in front of the variable. The coefficient indicates the factor by which you ______the

variable. The constant is the ______that is written by itself.

Example 1: Label the following expression with the terms “coefficient”, “constant”, and “variable”. /

Example 2: Identify the coefficient, variable, and constant in each expression.

coefficient
variable
constant

Before being able to solve word problems in algebra, you must be able to change words, phrases, and sentences into algebraic form.

Here is a list of commonly used words and the operations they correspond to in mathematics:

WORDS / PHRASES / ALGEBRAIC EXPRESSION
ADDITION / 5 more than
increased by 5
The sum of and 5
plus 5
5 added to
SUBTRACTION / 7 less than
decreased by 7
The difference of and 7
minus 7
7 subtracted from
WORDS / PHRASES / ALGEBRAIC EXPRESSION
MULTIPLICATION / 2 times
2 of
The product of 2 and
2 multiplied by
twice
DIVISION / divided by 4
The ratio of to 4
The quotient of and 4
per 4
EXPONENTS / The square of
The cube of

An expression is…

Example 3: Translate the following phrases into algebraic expressions.

a)Seven more than twice ______

b)Five less than the square of ______

c)2 more than the product of a number and 3______

d)5 times the sum of a number and 2______

e)The sum of 5 times a number and 2______

An equation is… / Here are some words that mean “is equal to”:
is
was
yields
is the same as / result in
gives
becomes

The words that translate to “ = ” divide the word sentence into the two sides of the equation.

The phrase beforethe “ = ” makes up the leftside of the equation.

The phrase afterthe “ = ” makes up the rightside of the equation.

Example 4: Translate the following sentences into algebraicequations.

a)The sum of a number and 3 is12. ______

b) Twice a number is the same as 8 more than the number . ______

c) Seven less than three times a number yields six more than the number. ______

d) The square of a number plus 8 gives six times the number. ______

e) Two times the sum of a number and 7 is 2 less than six times the number. ______

/ MATHEMATICS 8
CHAPTER 10 – SOLVING LINEAR EQUATIONS
MODELING / SOLVING (10.1) / Date ______

When a linear equation is graphed…

For simpler linear equations, we can solve them by inspection or by using algebra tilesor diagrams.

** When working with linear equations, what you do to one side, you must do the SAMEto the other side. **

Think of an equation as a scale and both sides are balanced. If you do something to one side, it tips. Therefore, you must do the same thing to the other side to balance it out again.

ALGEBRA TILES

/ represents a positive variable / / represents
/ represents a negative variable / / represents

Example 1: Solve the following.

a)

INSPECTION / ALGEBRA TILES

b)

INSPECTION / DIAGRAM

c)

INSPECTION / ALGEBRA TILES

d)

INSPECTION / ALGEBRA TILES

Solving by inspection or using algebra tiles can become more difficult if the numbers are very BIG or don’t work out nicely (eg decimals).

The best way to solve equations is to APPLY THE OPPOSITE OPERATION.

An opposite operation…

Operation / Opposite (or Inverse) Operation
Addition
Subtraction
Multiplication
Division

Example 2: Solve by applying the opposite operation. Check your answers to make sure they are correct!

a) / b)
c) / d)
e) / f)
g) / h)
Example 3: The average temperature in March in Vancouver is twice as warm as the average temperature in Toronto. If the temperature in Vancouver is 12C, what is the temperature in Toronto? / Example 4: Alex is making bead necklaces. She has 144 beads which she will use to make 9 necklaces. How many beads are on each necklace?
/ MATHEMATICS 8
CHAPTER 10 – SOLVING LINEAR EQUATIONS
MODELING / SOLVING (10.2) / Date ______

SOLVING EQUATIONS USING ALGEBRA TILES OR DIAGRAMS

Example 1: Solve the following usingbalance scale diagrams.

a)

b)

c)

Example 2: Solve the following using algebra tiles.

a)

b)

c)

SOLVING EQUATIONS BY APPLYING OPPOSITE OPERATIONS

Recall from last lesson:

ADDITION is the opposite of SUBTRACTION (and vice versa)

MULTIPLICATION is the opposite of DIVISION (and vice versa)

Example 3: What steps were done to “” to turn it into “”? What steps were done to “” to turn it back into “”?

Example 4: What steps are needed to turn each of the following back into “”?

a)
1.
2. / b)
1.
2. / c)
1.
2.

When solving an equation (or “isolating a variable”), follow the reverse order of operations (= reverse BEDMAS):

First, ADD and/or SUBTRACT,

then MULTIPLY and/or DIVIDE

REMEMBER: What you do to one side of the equation must also be done on the other side. (The balance must stay even or it will tip to one side!!!).

Example 5: Solve the following by applying the opposite operations. Check your answers.

a)

b)

c)

d)

Example 6: Anna is having friends over at her house. She collects $5 from everyone (to cover the food) except Alvin, who is only charged $2 (he brought over chips and dip). If she collected $57, how many friends (other than Alvin) did Anna invite?

/ MATHEMATICS 8
CHAPTER 10 – SOLVING LINEAR EQUATIONS
MODELING / SOLVING (10.3) / Date ______

SOLVING EQUATIONS USING ALGEBRA TILES

Example 1: Use algebra tiles to solve .

SOLVING EQUATIONS BY APPLYING OPPOSITE OPERATIONS

Recall from last lesson:

ADDITION is the opposite of SUBTRACTION (and vice versa)

MULTIPLICATION is the opposite of DIVISION (and vice versa)

When solving an equation (or “isolating a variable”), follow reverse BEDMAS:

First, ADD and/or SUBTRACT,

then MULTIPLY and/or DIVIDE

REMEMBER: What you do to one side of the equation must also be done on the other side. (The balance must stay even or it will tip to one side!!!).

Example 2: What steps are needed to turn each of the following back into “”?

a)
1.
2. / b)
1.
2. / c)
1.
2.

Example 3: Solve the following by applying the opposite operations. Check your answers.

a)

b)

c)

d)

Example 4:The cost for BJ to go to the monster truck rally is $3 less than one-third of his dad’s adult ticket. The cost of the child ticket is $6. How much is the adult ticket?

/ MATHEMATICS 8
CHAPTER 10 – SOLVING LINEAR EQUATIONS
MODELING / SOLVING (10.4) / Date ______

SOLVING EQUATIONS USING ALGEBRA TILES

Example 1: Use algebra tiles to solve .

SOLVING EQUATIONS BY APPLYING OPPOSITE OPERATIONS

Recall from the previous lessons:

ADDITION is the opposite of SUBTRACTION (and vice versa)

MULTIPLICATION is the opposite of DIVISION (and vice versa)

When solving an equation (or “isolating a variable”), follow reverse BEDMAS:

First, ADD and/or SUBTRACT,

then MULTIPLY and/or DIVIDE

REMEMBER: What you do to one side of the equation must also be done on the other side. (The balance must stay even or it will tip to one side!!!).

Example 2: What steps were done to “” to turn it into “”? What steps were done to “” to turn it back into “”?

Example 3: Solve the following by applying the opposite operations. Check your answers.

a)

b)

c)

SOLVING EQUATIONS BY USING THE DISTRIBUTIVE PROPERTY

Example 4: Solve the following using BEDMAS and the distributive property.

a)

METHOD 1: BEDMAS / METHOD 2: DISTRIBUTIVE PROPERTY

b)

METHOD 1: BEDMAS / METHOD 2: DISTRIBUTIVE PROPERTY
In general, the distributive property states that: /

Example 5: Solve each equation using the distributive property. Check your answers.

a)

b)

c)

Example 6: A square picture is framed with a border that is 3cm wide. If the total perimeter of the picture and frame is 480cm, what is the length of one side of the picture?