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Algebra
Solving Linear Equations
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What is an equation?
An equation must have an = sign. Both sides are the same
3x + 9 = 15
Left hand
side
Right hand
side
To balance an equation:
What you do to one side you must do to the other side.
An equation is solved
when you are have x
on its own on one
side e.g. x = 5
x
5
SLO
To solve one step equations
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Solving one step Equations
Steps to solving equations.
Step 1) Opposite on both sides
Step 2) Cancel one side
Step 3) Solve other side
Check your answer
Leave a couple centimetres above and below as there are other steps.
We will cover this later.
x + 9 = 15
_________
-9 -9
x
= 6
First, put the opposite (- 9) on both sides.
Second, cancel out the 9’s on the left side.
Third, combine the right hand side.
Check your answer: 6 + 9 = 15
K
7x = 3 x 7
7
K = 21
First, put the opposite on both sides.
Second, cancel out one side.
Third, combine the other side
Check your answer: 21 ÷ 7 = 3
x+9=5
_________
–9 –9
x=
-4
First, put the opposite (- 9) on both sides.
Second, cancel out the 9’s on the left side.
Third, combine the right hand side.
Check your answer: -4 + 9 = 5
-3x =___
15
___
-3
-3
x = -5
First, put the opposite on both sides.
Second, cancel out one side.
Third, combine the other side.
Check your answer
Your Turn:
Solve the following and show your workings
Question
x + 6 = 10
x–4=3
x+7=2
3x = -18
𝑋
3
=7
4x = 20
𝑋
4
= -2
Answer
4
7
-5
-6
21
5
-8
SLO
To solve 2 step equations
http://www.youtube.com/watch?v=juG-iIuTJQE&feature=relmfu
(2 step equations)
Examples of Two-Step Equations
2 Step Equations on a Balance
3x + 2
17
3x + 2 = 17
There are two operations to undo, but which comes first?
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Rule for Solving 2 Step Equations
3x + 2 = 17
The operation to undo first is the one that is furthest
away from x.
In the above case we do the opposite of +2 first.
Step 1) Opposite on both sides (furthest away first)
Step 2) Cancel one side
Step 3) Solve other side
Step 4) Repeat until left with x on its own
Check your answer
Ex. 1: Solve 3x + 2 = 17
3x + 2 = 17
-2 -2
3x
= 15
3
3
x=5
(Subtract 2 from both sides)
(Cancel one side & solve other side)
(Divide both sides by 3)
(Cancel one side & solve other side)
Check your answer: 3 x 5 + 2 = 17 correct
Ex. 2: Solve 2x + 7 = 19
2x + 7 = 19
-7 -7
2x
= 12
2
2
x=6
(Subtract 7 from both sides)
(Cancel one side & solve other side)
(Divide both sides by 2)
(Cancel one side & solve other side)
Check your answer: 2 x 6 + 7 = 19 correct
Your Turn:
Solve these and show your workings
1. 2x – 5 = 17 x = 11
2. 3y + 7 = 25
y=6
3. 5n – 2 = 38
n=8
4. 12b + 4 = 28
b=2
Questions to do from the books
2 step linear
Achieve
Gamma
CAT 1.2
Merit
P44 Ex4.01
P44 Ex4.02
P18 Q113 – 118, 127, 129
Excellence
SLO
To solve equations with x on both sides
http://www.youtube.com/watch?v=f15zA0PhSek (x both sides)
xxxxx
xx 15
Write the above balance as an equation
5x
=
2x
+
15
What is the fist step to solving this equation?
Take 2x
What would you have left if you take 2x from both sides?
3x = 15
The final step is to divide by 3 leaving x = 5
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Rule for Solving x on both sides
6x = 2x + 10
Remove x from one side
In the above example -2x from both sides
Clean up by having x on one side only (make 2 step equation): collect x
on side with most x’s
Step 1) Opposite on both sides (furthest away first)
Step 2) Cancel one side
Step 3) Solve other side
Step 4) Repeat until left with x on its own
Check your answer
Your Turn: solve the following
1) 4x = x + 9
2) 6x = 2x + 20
3) 7x = 3x + 8
4) 3x = x – 10
5) -2x = 4x - 60
x=3
x=5
x=2
x = -5
x = 10
Equations with x and numbers on both
sides
8
xxxxx
xx 20
5x + 8 = 2x + 20
Example: Solve
5x + 8 = 2x + 20
Subtract 2x from both sides
3x + 8 = 20
Subtract 8 from both sides
3x = 12
Divide both sides by 3
x=4
Your Turn: Solve the following
1) 2x + 7 = 4x - 13
2) 8x + 3 = 12x – 17
3) 5x – 9 = x + 15
4) 3x + 5 = 33 – x
x = 10
x=5
x=6
x=7
Questions to do from the books
x and numbers on both sides
Achieve
Merit
Gamma P45 Ex4.03 Q1 - 12 P45 Ex4.03 Q13 - 18
CAT 1.2
P18 Q119 – 122,
131, 132
Excellence
SLO
Solving equations with brackets
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Rule for solving equations with brackets
6(x + 2) = 30
Expand brackets first
In the above example this will leave 6x + 12 = 30
Expand brackets
Clean up by having x on one side only (make 2 step equation)
Step 1) Opposite on both sides (furthest away first)
Step 2) Cancel one side
Step 3) Solve other side
Step 4) Repeat until left with x on its own
Check your answer
Example 1
multiply out the
brackets:
subtract 6x from
both sides:
3(2x + 4) = 10x
6x + 12 = 10x
-6x
-6x
12 = 4x
divide both sides by 4:
÷4 ÷4
3=x
x=3
Can you
explain why
we subtract
6x not
subtract 12?
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Example 2
2(3x – 5) = 4x
To solve this
multiply out the brackets:
6x –10 = 4x
+10
add 10 to both sides:
+10
6x = 4x + 10
- 4x - 4x
subtract 4x from both sides:
2x = 10
÷2 ÷2
divide both sides by 2:
x=5
We add 10 and
not subtract 6x
as there are
more x’s on
the left
Your Turn
Question
6(x – 3) = 12
3(2x – 6) = 30
4(3x – 1) = 8
7(3x – 11) = 70
Answer
5
8
1
7
Questions to do from the books
brackets
Achieve
Gamma P46 Ex 4.04
CAT 1.2
P18 Q123 – 126, 128, 130
Merit
Excellence
SLO
Merit: Solving equations with fractions
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Solving equations with fractions
𝑥+3
5
=2
Multiply both sides by denominator(s).
In the above example, multiply both sides by 5. This will
leave x + 3 = 10
Get rid of fractions by multiplying everything by denominator(s)
Expand brackets
Clean up by having x on one side only (make 2 step equation)
Step 1) Opposite on both sides (furthest away first)
Step 2) Cancel one side
Step 3) Solve other side
Step 4) Repeat until left with x on its own
Check your answer
Solving equations with fractions
3
x
4
–5=9–x
We can remove the 4 from the denominator by
multiplying both sides of the equation by 4.
3
4( x – 5) = 4(9 – x)
4
expand the brackets: 3x – 20 = 36 – 4x
add 4x to both sides: 7x – 20 = 36
7x = 56
add 20 to both sides:
x=8
divide both sides by 7:
Example
2x + 7
=x–1
5
To remove the 5 from the denominator we multiply both sides
of the equation by 5.
2x + 7 = 5(x – 1)
expand the brackets:
2x + 7 = 5x – 5
5x – 5 = 2x + 7
add 5 to both sides:
5x = 2x + 12
subtract 2x from both sides:
3x = 12
divide both sides by 3:
x=4
swap sides:
Example: Solve
3x + 2 = 11 – x
4
In this case we must start by multiplying both sides of the
equation by 4.
3x + 2 = 4(11 – x)
Multiply out the brackets:
3x + 2 = 44 – 4x
Add 4x to both sides:
7x + 2 = 44
Subtract 2 from both sides:
Divide both sides by 7:
7x = 42
x=6
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Example 1:
2
1
x
=
x
+
1
3
2
Get rid of ‘over two’, multiply both sides by 2.
2
1
2( x) =2( x +
3
2
4
𝑥=x+2
3
1)
expand the brackets:
Get rid of ‘over 3’, multiply both sides by 3.
12
𝑥
3
=
6
x
2
+6
4x = 3x + 6
x=6
Simplify fractions.
subtracting 3x from both sides:
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Example 1 (Quicker method):
2
1
x
=
x
+
1
3
2
Step 1: Get rid of factions by multiplying both sides by 2 and
then by 3. The quick way is to multiply both sides by
multiplying by the lowest common multiple of the two
denominators.
E.g. What is the lowest common multiple of 3 and 2?
The lowest common multiple of 3 and 2 is 6.
2
6(3x
)=
12
𝑥
3
1
6(2x
+ 1)
6
= 2x + 6
expand the brackets:
subtracting 3x from both sides:
4x = 3x + 6
x=6
Example 2:
5x – 3
2x – 1
=
4
3
Either multiply both sides by 3 and then by 4 or the quick way:
multiply by 12
4
3
12(5x – 3)
12(2x – 1)
=
41
31
which simplifies to:
expand the brackets:
3(5x – 3) = 4(2x – 1)
15x – 9 = 8x – 4
add 9 to both sides:
15x = 8x + 5
7x = 5
5
divide both sides by 7:
x=
7
Although this answer could be written as a rounded decimal,
it is more exact left as a fraction.
subtract 8x from both sides:
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We have seen that
simplifies to
Cross multiplication
5x – 3
2x – 1
=
4
3
3(5x – 3) = 4(2x – 1)
How could we perform this simplification in one step?
Doing this in one step in often called cross multiplication.
Take care: Cross multiplication only works with a fraction
either side of the equal sign. You can not use it in the
following example:
3𝑥+5
8
4 + 6 = 𝑥 −6
This 4 prevents us using the cross multiplication method.
Sometimes the unknowns appear in the denominator.
For example,
4
5
=
(x + 3)
(3x – 5)
In this example, we can use cross multiplication to give:
4(3x – 5) = 5(x + 3)
expand the brackets:
subtract 5x from both sides:
add 20 to both sides:
divide both sides by 7:
12x – 20 = 5x + 15
7x – 20 = 15
7x = 35
x=5
Press play to show cross multiplication
Equivalent equations
Your Turn: Solve
Question
𝑥+7
3
=5
𝑥+3
3𝑥 + 1
=
2
4
2
𝑥 + 7 = 17
3
3
2
𝑥+7= 𝑥+8
4
3
Answer
x=8
x=5
x = 15
x = 12
Questions to do from the books
fractions
Achieve
Merit
Gamma
P54 Ex4.08/9/10
CAT 1.2
P19 Q133 – 138
Excellence