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Solving equations
When solving an equation we should keep algebra in every step of our method.
When we 'undo' we do the inverse (opposite) e.g. 'Add' becomes 'Subtract'.
On the RHS always start with what was already there.
Removing only one step at a time means everything else remains the same and in the same
place.
A. SOLVING THREE TERM EQUATIONS
a.
3x + 2 = 17
b.
12 + 4x = 32
3x = 15
x =
4x = 20
15
3
-2x = -8
x =
4
x =5
5x – 6 = 15
21 = 7x
21
21
5
7
x = 4.2
=x
3 =x
−8
−2
x =4
20 = 7x - 1
e.
5x = 21
x =
20 – 2x = 12
Don’t lose
the sign
20
x =
x =5
d.
c.
f.
7x - 1 = 20
7x = 21
Completely
swap sides to
bring x onto
the LHS
x =
21
7
x =3
B. Solving equations with brackets
a.
3(2x – 1) = 27
b.
6x - 3 = 27
Expand
the
brackets
2(4x + 9) = 6
8x + 18 = 6
6x = 30
x =
8x = -12
30
x =
6
x =5
−12
Expand
the
brackets
2(4x + 9) = 6
Divide by
the
coefficient
4x + 9 =
2
4x + 9 = 3
4x = -6
8
x = -1.5
6
x =
Choose the
method you find
easiest
−6
4
=1.5
C. Solving equations with fractions
Only part of LHS is divided by 3.
Either lose the ‘odd bit’ first
or multiply every term by 3
Be careful. Look carefully to see what is being divided
All LHS is divided by 2 so multiply RHS by 2
a.
(2) 3x + 4 (2)
2
= 11
3x + 4 = 22
3x = 18
x =
b.
5x
3
-1 =9
5x
(3)
(3) 5x(3)
3
-1 =9
= 10
5x - 3 = 27
5x = 30
5x = 30
3
18
x =
3
x =6
30
5
x =6
x =
30
5
x =6
Choose the method you find easiest
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Solving equations
D. Solving equations with unkowns on both sides
Collect all the letters onto one side and the numbers onto the other.
a.
6x - 7 = 2x + 5
b.
4x - 7 = 5
7x - 14 = 5x - 4
5x + 4 = 20 – 3x
2x - 14 = -4
8x + 4 = 20
2x = 10
8x = 16
4x = 12
x =
c.
12
x =
4
x =3
10
x =
2
x =5
16
2
x =2
You sometimes have various options to solve an equation
d.
4x + 5 = 7x - 4
-3x + 5 = -4
-3x = -9
x =
−9
−3
4x + 5 = 7x - 4
Completely
swap sides to
put the largest
x term on the
LHS
4x + 5 = 7x - 4
7x - 4 = 4x + 5
3x - 4 = 5
3x = 9
x =3
x =
5 = 3x - 4
Taking the
letters to the
RHS
9 = 3x
9
3
9
=x
3 =x
3
x =3
e.
23 – 2x = 6x - 5
6x – 5 = 23 -2x
23 – 8x = -5
8x - 5 = 23
-8x = -28
x =
23 – 2x = 6x - 5
23 = 8x - 5
8x = 28
−28
x =
−8
x = 3.5
28 = 8x
28
28
8
8
x = 3.5
=x
3.5 = x
Choose the method you find easiest
f.
2(3x – 8) = 5(2x – 2)
6x - 16 = 10x - 10
10x - 10
= 6x –
16
4x – 10 = -16
4x = -6
x =
−6
4
x = -1.5
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Expand the
brackets
Swap
sides
g)
2(4x + 25) = 4(5 – x)
8x + 50 = 20 – 4x
12x + 50 = 20
Don’t skimp on method
Work slowly
Work carefully
Work accurately
Just follow the rules and
it will work!
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12x = -30
x =
−30
12
6
x = -2
12
x = -2.5
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Solving equations
E. Solving harder equations with fractions
To solve equations with fractions it is likely that we will need to find a common denominator
and cross multiply.
4y + 8
5
a.
c.
=
5(3y - 5)
4x x
−
3 2
2(4x) − 3(x)
6
8x − 3x
6
5x
6
5x
=
10
=
10
=
10
=
10
=
60
4y + 8
=
4y + 8
=
15y - 25
-11y + 8
=
-25
-11y
=
-33
y
=
−33
−11
x
=
60
5
y
=
3
x
=
12
=
10
=
4x − 3 2x + 1
+
5
3
3(4x − 3) + 5(2x + 1)
15
12x − 9 + 10x + 5
15
22x − 4
15
15 is the
common
22x - 4
denominator
b.
3y - 5
6 is the
common
denominator
3x + 8
5
=
x + 12
4
10
4(3x + 8 )
=
5(x + 12)
=
10
12x + 32
=
5x + 60
=
10
7x + 32
=
60
=
150
7x
=
28
22x
=
154
x
=
x
=
154
22
28
7
x
=
4
x
=
7
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d.
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