Download Tutorial 3e - C on T ech Math : : An application

Solving Problems
with Linear Equations in 1 Variable
Tutorial 3e
Introduction:
Before building bridges, skyscrapers, or rollercoasters, engineers and architects make models.
Not all models are made of wood, metal, or plastic.
Mathematical models are made of equations.
These equations can predict how cables, beams,
and coaster cars will perform in real life.
You can solve many math problems by writing
equations to model a situation.
Introduction:
Sometimes a problem cannot be solved directly, or with
the previous problem solving methods explored in
tutorial 3a, instead algebraic equations are necessary.
Often the hardest part of problem solving is,
understanding what is being asked. Therefore, a helpful
tool is to use the following five-step strategy for
breaking down the problem.
Five-Step Strategy:
1.
Read the problem carefully and decide what
numbers are being asked for.
2.
Choose and define your variable. Then use it with
the given facts to represent the number(s)
described in the problem. Making a drawing may
be helpful.
3.
Write an equation based on the given facts.
4.
Solve the equation and find the required numbers.
5.
Check your answer.
Review 5-Steps: 1.
2.
Example 1
3.
4.
5.
Read the problem carefully and decide what numbers are being
asked for.
Choose and define your variable and use it with the given facts
to represent the number(s) described in the problem. Making a
drawing may be helpful.
Write an equation based on the given facts.
Solve the equation and find the required numbers.
Check your answer.
Maria bought a hit CD. The cost of the CD is $18. Her friend
bought a cassette of the same music. If the CD cost twice as
much as the cassette, what was the cost of the cassette?
Step 1: Read the problem carefully and decide what number(s)
are being asked for.
As you read, look for key words or phrases such as sum,
difference, product, is less than, or twice as much as.
Sometimes it is helpful to underline key phrases (see above).
• This problem asks for the number of dollars in the cost
of the cassette.
Review 5-Steps: 1.
2.
Example 1
3.
4.
5.
Read the problem carefully and decide what numbers are being
asked for.
Choose and define your variable and use it with the given facts
to represent the number(s) described in the problem. Making a
drawing may be helpful.
Write an equation based on the given facts.
Solve the equation and find the required numbers.
Check your answer.
Maria bought a hit CD. The cost of the CD is $18. Her friend
bought a cassette of the same music. If the CD cost twice as
much as the cassette, what was the cost of the cassette?
Step 2: Choose and define your variable and use it with the
given facts to represent the number(s) described in the
problem. Making a drawing may be helpful.
Let n = cost of the cassette in dollars.
Next try to write the other values discussed in the
problem in terms of n (see above).
Let 2n = cost of the CD in dollars.
Review 5-Steps: 1.
2.
Example 1
3.
4.
5.
Read the problem carefully and decide what numbers are being
asked for.
Choose and define your variable and use it with the given facts
to represent the number(s) described in the problem. Making a
drawing may be helpful.
Write an equation based on the given facts.
Solve the equation and find the required numbers.
Check your answer.
Maria bought a hit CD. The cost of the CD is $18. Her friend
bought a cassette of the same music. If the CD cost twice as
much as the cassette, what was the cost of the cassette?
Step 2: Let n = cost of the cassette in dollars.
Let 2n = cost of the CD in dollars.
Step 3: Write an equation based on the given facts.
The information in the 2nd sentence gives us the information we
need to write an equation. To write the equation, we will simply
translate the sentence into math symbols
Solve the following equation:
The cost of the CD is $18.
2n = 18
2n
= 18
Review 5-Steps: 1.
2.
Example 1



3.
4.
5.
Read the problem carefully and decide what numbers are being
asked for.
Choose and define your variable and use it with the given facts
to represent the number(s) described in the problem. Making a
drawing may be helpful.
Write an equation based on the given facts.
Solve the equation and find the required numbers.
Check your answer.
Maria bought a hit CD. The cost of the CD is $18. Her friend
bought a cassette of the same music. If the CD cost twice as
much as the cassette, what was the cost of the cassette?
Step 3: Write an equation based on the given facts.
Step 4: Solve the equation and find the required numbers.
Solve the following equation:
Think to yourself: What is being done to
12n = 18 9
the variable (n)?
A 2 is being multiplied to the variable (n).
2 1
12
Division undoes multiplication, therefore
n = 9
you should divide a 2 to the left side to get
n alone on that side.
Solve the following equation:
However, whatever you do to one side of an
2n = 18
equation you must also do to the other side.
Review 5-Steps: 1.
2.
Example 1
Review:
3.
4.
5.
Read the problem carefully and decide what numbers are being
asked for.
Choose and define your variable and use it with the given facts
to represent the number(s) described in the problem. Making a
drawing may be helpful.
Write an equation based on the given facts.
Solve the equation and find the required numbers.
Check your answer.
Maria bought a hit CD. The cost of the CD is $18. Her friend
bought a cassette of the same music. If the CD cost twice as
much as the cassette, what was the cost of the cassette?
Step 2: Let n = cost of the cassette in dollars.
Let 2n = cost of the CD in dollars.
Step 3: Write an equation based on the given facts.
Step 4: Solve the equation and find the required numbers.
Step 5: Check the answer.
Solve the following equation:
12n = 18 9
Check the results with the statements
Answer:
of the problem:
2 1
12
• Is the cost of the CD ($18) twice the cost
The cassette
cost
$9.00
n
=
9
of the cassette? 2n = 18
2 • 9 = 18
18 = 18 Yes 
Example 2
Jim is 19 years old. He was 15 when his brother Bill was
born. How old is Bill?
Step 2: Choose and define your variable and use it with the given facts
to represent the number(s) described in the problem.
Define your variable: Let b = Bill’s age.
Let b + 15 = Jim’s age.
Step 3: Write an equation:
Relate:
is
19 years old.
Jim
Write:
=
b + 15
Step 4: Solve the equation:
Solve:
b + 15 = 19
19
Example 2
Let b = Bill’s age
Let b + 15 = Jim’s age.
Jim is 19 years old. He was 15 when his brother Bill was
born. How old is Bill?
Think to yourself: What is being Step 4:
done to the variable (b)?
 A 15 is being added to the
variable (b). Subtraction undoes
addition therefore you should
subtract a 15 on the left to get b Step 5:
alone on one side.
 However, whatever you do to one
Step
4: ofSolve
the equation:
side
an equation
you must also
Solve:
do to the
other side.b + 15 = 19Answer:

Solve the equation:
b + 15 = 19
-15 -15
b = 4
Check the answer.
b + 15 = 19
4 + 15 = 19
19 = 19 
Bill is 4 years old.