Applications of Equations

1
Applications of Equations
Lesson 1.2.6
2
Lesson 1.2.6
Applications of Equations
California Standards Algebra and Functions
4.1 Solve two-step linear equations and
inequalities in one variable over the rational
numbers, interpret the solution or solutions in
the context from which they arose, and verify the
reasonableness of the results. Mathematical
Reasoning 2.1 Use estimation to verify the
reasonableness of calculated results.
What it means for you Youll see how to use
equations to help solve real-life math problems,
and how to check if your answer is sensible.

• Key Words
• model
• check
• reasonable
• sensible

3
Lesson 1.2.6
Applications of Equations
Equations can be really useful in helping you to
understand real-life situations.
Writing an equation can help you sort out the
information contained in a word problem and turn
it into a number problem.
4
Lesson 1.2.6
Applications of Equations
Equations Can Describe Real-Life Situations
An equation can help you to model a real-life
situation to describe it in math terms.
You can use an equation to help you describe
almost any situation that involves numbers and
unknown numbers.
5
Lesson 1.2.6
Applications of Equations

• For example
• Youve just had your car repaired. The bill was
  280.
• You know the parts cost 120.
• You know the mechanic charges labor at 40 per
  hour.
• You want to know how long the mechanic worked on
  your car you need an equation.

Let h number of hours worked by mechanic.
1. Choose a variable.
2. Write an equation.
40h 120 280
3. Solve the equation.
Subtract 120 from both sides
The amount billed
Cost of labor
40h 160
Cost of parts
So the mechanic must have worked on your car for
4 hours.
Divide both sides by 40
h 4
6
Lesson 1.2.6
Applications of Equations
Example 1
At the school supply store, Mr. Ellis bought a
notebook costing 3, and six pens. He spent 15
in total. Find the price of one pen, p.
Solution

• First write out the information you have
• Total spent 15
• Cost of notebook 3
• Cost of six pens 6p

You know that six pens and the notebook cost a
total of 15. So you can write an equation with
the cost of each of the items bought on one side,
and the total spent on the other.
Solution follows
Solution continues
7
Lesson 1.2.6
Applications of Equations
Example 1
At the school supply store, Mr. Ellis bought a
notebook costing 3 and six pens. He spent 15
in total. Find the price of one pen, p.
Solution (continued)
6p 3 15
The total cost
Cost of pens
Cost of notebook
Now you have a two-step equation. You can find
the cost of one pen by solving it.
Solution continues
8
Lesson 1.2.6
Applications of Equations
Example 1
At the school supply store, Mr. Ellis bought a
notebook costing 3 and six pens. He spent 15
in total. Find the price of one pen, p.
Solution (continued)
6p 3 15
Write out the equation
6p 12
Subtract 3 from both sides
p 2
Divide both sides by 6
One pen costs 2.
9
Lesson 1.2.6
Applications of Equations
Guided Practice
Write an equation to describe each of the
situations in Exercises 13. Then solve it to
find the value of the variable. 1. Emily is seven
years older than Ariela. The sum of their ages
is 45. How old is Ariela? 2. A sale rack at a
store has shirts for 9 each. Raul has 50 and
a coupon for 4 off any purchase. How many
shirts can he buy? 3. The price for renting bikes
is 15 for half a day, then 3 for each
additional hour. How many hours longer than half
a day can you keep a bike if you have 24?
A A 7 45, A 19
9S 4 50, S 6
15 3h 24, h 3
Solution follows
10
Lesson 1.2.6
Applications of Equations
You Need to Check That Your Answer is Reasonable
When youve solved an equation that describes a
real-life problem, you need to look at your
answer carefully and see if it is reasonable.
Here are two important things to think about

1. Does your answer make sense?

1. Is your answer about the right size?

11
Lesson 1.2.6
Applications of Equations
1) Does Your Answer Make Sense?
You must always check that the answer makes sense
in the context of the question. For example
An orchard charges 1.10 for a pound of apples.
You have 8.25. How many pounds of apples can
you buy?
Set up an equation to describe the
problem Number of pounds 8.25 1.10 7.5
This is a reasonable answer as the orchard will
happily sell you half a pound of apples.
12
Lesson 1.2.6
Applications of Equations
But look what happens if you change the problem
slightly.
A store charges 1.10 for a bag of apples. You
have 8.25. How many bags of apples can you buy?
Set up an equation to describe the
problem Number of bags 8.25 1.10 7.5
This is not a reasonable answer the store
wouldnt sell you half a bag of apples. You
could only buy 7 bags.
13
Lesson 1.2.6
Applications of Equations
2) Is Your Answer About the Right Size?
The size of your answer has to make sense in
relation to the question that is being asked.
For example

1. If youre finding the height of a mountain, and
   your answer is 5 feet, its not reasonable.

1. If youre finding the height of a person, and
   your answer is 5000 feet, thats not reasonable
   either.

If the size of your answer doesnt seem
reasonable then its really important to go back
and check your work to see if youve made an
error somewhere.
14
Lesson 1.2.6
Applications of Equations
Example 2
Kea is going to walk 1.5 miles at a steady speed
of 3 miles per hour. She works out how long it
will take using the work shown. Is her answer
reasonable?
Solution
Given that Keas walk is only 1.5 miles long and
she walks at 3 mi/h, 4.5 hours is not a
reasonable answer it is much too long.
(Kea multiplied the distance of the walk by her
speed. She should divide the distance by the
speed instead Time 1.5 3 0.5 hours.)
Solution follows
15
Lesson 1.2.6
Applications of Equations
Guided Practice
4. Pete is buying trading cards. One card costs
20. He says 10 cards will cost 20. Is this a
sensible answer? Explain why or why not. 5. Six
friends earn 87 washing cars. How much will
each one get if they split it evenly? Is your
answer reasonable in the context of the question?
This is not reasonable his answer is much too
big. 10 cards will cost 2.
Each will get 14.50. This is reasonable, as
they can get fractions of dollars each (and it
seems about the right size).
Solution follows
16
Lesson 1.2.6
Applications of Equations
Guided Practice
6. A yard has a 150-foot perimeter. Fencing is
sold in 40-foot rolls. Write an equation to
describe the number of rolls, n, you need to buy
to fence the yard. Solve the equation. Is your
answer reasonable in the context of the
question? 7. Ana is as tall as T.J., who is
174 cm tall. Write an equation to describe
Anas height, A. Solve it. Is the size of your
answer reasonable?
40n 150, n 3.75. This isnt reasonable you
couldnt buy part of a roll. You would have to
buy 4.
Solution follows
17
Lesson 1.2.6
Applications of Equations
Independent Practice
Write an equation to describe each situation in
Exercises 12, and solve the equation to answer
the question. 1. Don has spent 474 ordering
sticks for his hockey team. A stick costs 50.
Shipping costs 24. How many did he buy? 2.
Tiana is saving up to buy a fishing rod. The
rod costs 99 with tax. She already has 27,
and can afford to save another 12 each week.
How long will it take her to save enough for the
rod?
50x 24 474, 9 sticks.
27 12x 99, 6 weeks.
Solution follows
18
Lesson 1.2.6
Applications of Equations
Independent Practice
3. Joy went to the fabric store to buy ribbon.
She got f feet, and spent 5. The ribbon cost
80 a foot. Write an equation to describe how
much she got. Solve it. Is your answer
reasonable in the context of the question? 4.
Mike is asked to multiply 5 by . He says the
answer is 10. Is this reasonable in the context
of the question? Explain why or why not.
0.8f 5, f 6.25 feet. This is reasonable the
store is likely to be happy to sell her part of a
foot of ribbon.
This isnt reasonable when you multiply a number
by a fraction less than 1, the answer should be
smaller than the original number.
Solution follows
19
Lesson 1.2.6
Applications of Equations
Independent Practice
5. Two friends run a dog walking service, each
walking the same number of dogs. Write and solve
an equation to show how many dogs, d, each friend
walks if they walk nine dogs between them. Is
your answer reasonable?
2d 9, d 4.5. This isnt reasonable. You
couldnt walk part of a dog.
Solution follows
20
Lesson 1.2.6
Applications of Equations
Round Up
Equations can help you to understand situations.
They can also help you to describe a real-life
math problem involving an unknown number and come
up with a solution.
But dont forget to always think carefully about
whether the answer is a reasonable one in
relation to the question.