Problem-Solving with Linear Equations
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18 February 2025
Problem solving using linear
equations
LO: To create equations from word problems
in order to solve them.
Problem solving with linear equations
When problems are solved using algebra we follow these
steps:
Step 1 :
Determine what the unknown quantity is and
use a letter or symbol to represent it.
Step 2 :
Decide which operations are involved.
Step 3 :
Translate the problem into an equation.
Step 4 :
Solve the equation by isolating the unknown.
Step 5 :
Check that your solution satisfies the original
problem.
Step 6 :
Write your answer in sentence form.
Constructing an equation
Example 1
I’m thinking of a number.
When I subtract 9 from the number and double it, I
get the same answer as dividing the number by 5.
What number am I thinking of?
The unknown is a number.
n
– 9
=
2( )
Step 1 :
Determine what the unknown quantity is and use a letter or symbol to
represent it.
Let’s call the unknown number
n
.
Step 2 :
Decide which operations are involved.
Step 3 :
Translate the problem into an equation.
Step 4 :
Solve the equation by isolating the unknown.
Solving the equation
We can solve this equation by performing the same operations
on both sides,
multiply both side by 5:
10(
n
– 9) =
n
add 90 to both sides:
10
n
=
n
+ 90
We can check the solution by substituting it back into the
original equation:
2
× (10 – 9)
=
10
÷
5
expand the bracket:
10
n
– 90 =
n
subtract
n
from both sides:
9
n
= 90
divide both sides by 9:
n
= 10
So, the number is 10
Step 5 :
Check that your solution satisfies the original problem.
Step 6 :
Write your answer in sentence form.
2
× (1)
=
`2
The sum of the ages of Ben and his daughter, Alice, is 66
years. In one year’s time Ben will be three times Alice’s age.
How old is Alice now?
Let’s call current Alice’s age
a
.
Looking at the first piece of information both ages add up to 66
66 –
a.
67 –
a
=
3(
a
+ 1)
Alice
Ben
Now
1 year time
a
66-
a
a
+ 1
67-
a
Using equations to solve problems
we can write Ben’s current age as
a + b =
66
In one year’s time Ben’s age will be
three times Alice’s age
Alice is 16, so Ben is 50. In one year Alice will be 17 and Ben
will be 51.
Solving the equation:
67 –
a
= 3(
a
+ 1)
Expand the brackets:
67 –
a
= 3
a
+ 3
subtract 3 from each side:
64 –
a
= 3
a
add
a
to both sides:
64 =
divide both sides by 4:
a
= 16
51 is three times 17, so our solution is correct.
4
a
So, Alice’s present age is 16 years
Using equations to solve problems
The sum of the ages of Ben and his daughter, Alice, is 66
years. In one year’s time Ben will be three times Alice’s age.
How old is Alice now?
Carl has only 20 cent coins and 50 cent coins in his wallet. He
has three more 50 cent coins than 20 cent coins, and their total
value is $2.90.
How many 20 cent coins does Carl have?
If Carl has
x,
20 cent coins
20
x
=
+ 50(
x
+ 3)
20 cent
50 cent
Number
Value
x
x
+ 3
20
x
cents
50(
x
+ 3) cents
Coin
290
Using equations to solve problems
then he has (
x
+ 3) 50 cent coins
Carl has 20 cent coins and 50 cents coins
How many 20 cent coins he has?
x
The total value of the coins is $2.90 or 290 cents
Carl has only 20 cent coins and 50 cent coins in his wallet. He
has three more 50 cent coins than 20 cent coins, and their total
value is $2.90.
How many 20 cent coins does Carl have?
20
x
=
+ 50(
x
+ 3)
290
20
x
=
+ 50
x
+ 150
290
=
70
x
+ 150
290
=
70
x
140
=
x
2
So, Carl has two 20 cent coins
Using equations to solve problems
Expand the brackets:
subtract 150 from both sides:
divide both sides by 70:
Simplifying:
Grandad is 4 times older than Jim
Jim
Write an expression for the sum of
their ages?
To solve this problem, we need to start by
letting Jim’s age be something
and then work out Grandad’s age in terms of
j
:
4
j
If their ages add up to 75
+
75
say
j
Jim’s
Grandad
j
4
j
=
5j = 75
j = 15
15
60
Using equations to solve problems
The sum of their ages is
j + 4j
So, Chris has 41 cards,
Three friends have 200 game cards between them. Joe has
more
cards than Chris. Kay has
How many cards does each friend have?
Let the number of cards which Chris has be
x
This means that:
x
+ 36
cards
Kay has
2
x
cards
So, the total number of cards between them is:
x
+
This gives:
4
x
+ 36 = 200
4
x
= 200 – 36
x
= 41
Joe has 77 cards
and Kay has 82 cards.
Using equations to solve problems
x
+ 36
2
x
= 4
x
+ 36
+
36
more
twice as many cards as Chris
.
36
more
Joe has
4
x
= 164
twice as many cards as Chris.
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Learn how to create and solve equations from word problems involving linear equations. Follow six steps: define the unknown, determine operations, translate into an equation, solve by isolating the unknown, check the solution, and present the answer in sentence form. Step through examples of constructing and solving equations, such as finding a number based on given conditions and determining ages using algebraic equations. Understand the process of problem-solving with linear equations with detailed explanations and illustrations.
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Presentation Transcript
18 February 2025 Problem solving using linear equations LO: To create equations from word problems in order to solve them. www.mathssupport.org
Problem solving with linear equations When problems are solved using algebra we follow these steps: Step 1 : Determine what the unknown quantity is and use a letter or symbol to represent it. Step 2 : Decide which operations are involved. Step 3 : Translate the problem into an equation. Step 4 : Solve the equation by isolating the unknown. Step 5 : Check that your solution satisfies the original problem. Step 6 : Write your answer in sentence form. www.mathssupport.org www.mathssupport.org
Constructing an equation Example 1 I m thinking of a number. When I subtract 9 from the number and double it, I get the same answer as dividing the number by 5. What number am I thinking of? Step 1 : Determine what the unknown quantity is and use a letter or symbol to represent it. The unknown is a number. Let s call the unknown number n. Step 2 : Decide which operations are involved. Step 3 : Translate the problem into an equation. n 5 n 9 = 2( ) Step 4 : Solve the equation by isolating the unknown. www.mathssupport.org
Solving the equation We can solve this equation by performing the same operations on both sides, 2(n 9) = n 5 10(n 9) = n 10n 90 = n multiply both side by 5: expand the bracket: 10n = n + 90 9n = 90 add 90 to both sides: subtract n from both sides: n = 10 divide both sides by 9: Step 5 : Check that your solution satisfies the original problem. We can check the solution by substituting it back into the original equation: 2 (10 9) = 10 5 2 (1) = `2 Step 6 : Write your answer in sentence form. So, the number is 10 www.mathssupport.org
Using equations to solve problems The sum of the ages of Ben and his daughter, Alice, is 66 years. In one year s time Ben will be three times Alice s age. How old is Alice now? Now a 66- a 1 year time a + 1 67- a Alice Ben Let s call current Alice s age a. Looking at the first piece of information both ages add up to 66 a + b = 66 66 a. we can write Ben s current age as In one year s time Ben s age will be three times Alice s age 67 a = 3(a + 1) www.mathssupport.org
Using equations to solve problems The sum of the ages of Ben and his daughter, Alice, is 66 years. In one year s time Ben will be three times Alice s age. How old is Alice now? 67 a = 3(a + 1) 67 a = 3a + 3 64 a = 3a 64 = a = 16 Solving the equation: Expand the brackets: subtract 3 from each side: add a to both sides: 4a divide both sides by 4: Alice is 16, so Ben is 50. In one year Alice will be 17 and Ben will be 51. 51 is three times 17, so our solution is correct. So, Alice s present age is 16 years www.mathssupport.org
Using equations to solve problems Carl has only 20 cent coins and 50 cent coins in his wallet. He has three more 50 cent coins than 20 cent coins, and their total value is $2.90. How many 20 cent coins does Carl have? Carl has 20 cent coins and 50 cents coins How many 20 cent coins he has? x If Carl has x, 20 cent coins then he has (x + 3) 50 cent coins Value 20x cents 50(x + 3) cents Coin Number x x + 3 20 cent 50 cent The total value of the coins is $2.90 or 290 cents 290 20x + 50(x + 3) = www.mathssupport.org
Using equations to solve problems Carl has only 20 cent coins and 50 cent coins in his wallet. He has three more 50 cent coins than 20 cent coins, and their total value is $2.90. How many 20 cent coins does Carl have? 290 20x + 50(x + 3) = 290 20x + 50x + 150 = Expand the brackets: 290 70x + 150 = Simplifying: subtract 150 from both sides: 140 70x = 2 x = divide both sides by 70: So, Carl has two 20 cent coins www.mathssupport.org
Using equations to solve problems Jim s Grandad Jim Grandad is 4 times older than Jim 15 60 Write an expression for the sum of their ages? To solve this problem, we need to start by letting Jim s age be something and then work out Grandad s age in terms of j: j say j 4j 4j The sum of their ages is + If their ages add up to 75 j + 4j = 75 5j = 75 j = 15 www.mathssupport.org
Using equations to solve problems Three friends have 200 game cards between them. Joe has more cards than Chris. Kay has more more 36 36 twice as many cards as Chris. twice as many cards as Chris. How many cards does each friend have? Let the number of cards which Chris has bex This means that: Kay has 2 x cards So, the total number of cards between them is: x + 36 cards x + 36 2 x Joe has This gives:4 x + 36 = 200 x + = 4 x + 36 + 4 x = 200 36 4x = 164 x = 41 So, Chris has 41 cards, Joe has 77 cards and Kay has 82 cards. www.mathssupport.org
Thank you for using resources from A close up of a cage Description automatically generated For more resources visit our website https://www.mathssupport.org If you have a special request, drop us an email info@mathssupport.org Get 20% off in your next purchase from our website, just use this code when checkout: MSUPPORT_20 www.mathssupport.org
