Better Math – Numeracy Basics
Algebra - Rearranging and Solving Linear Equations
Key
On screen content
Narration – voice-over
Activity – Under the Activities heading of the online program
Introduction
This topic will cover:
• the definition of a linear equation;
• techniques for rearranging linear equations; and
• how to rearrange a linear equation in order to solve for the value of a variable.
Welcome to Algebra.
This topic will cover:
•
•
•
the definition of a linear equation;
techniques for rearranging linear equations; and
how to rearrange a linear equation in order to solve for the value of a variable.
What is a Linear Equation?
Recall that an algebraic equation is an algebraic expression that also contains an equals (=) sign. For
example:
Algebraic Expressions (have no
equals signs)
5+d
3n + 1
6–f
a/2
Algebraic Equations (have equals
signs)
5 + d = 10
3n + 1 = 4
6–f=5
a/2 = 3
A particular kind of algebraic equation is a linear equation, or first degree equation. This is an equation
that only has variables raised to the first power, for example a – 3 = 5, as opposed to a2 – 3 = 5.
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Linear Equations
When a linear equation only contains only variable, then we can determine the value of the variable
and hence solve the equation. How to do this will be covered in this topic.
Recall that an algebraic equation is just an algebraic expression that also contains an equals sign.
Compare, for example, the algebraic expressions and algebraic equations listed in the table.
The first algebraic expression is 5 plus d, and we can turn this into an equation by simply setting this
equal to something. For example, 5 plus d equals 10 is an algebraic equation.
The second algebraic expression is 3n plus 1, which becomes an algebraic equation when it is set
equal to 4.
The third algebraic expression is 6 subtract f, which becomes an algebraic equation when it is set
equal to 5.
Finally, the algebraic expression a divided by 2 can be set equal to 3, to give the algebraic equation a
divided by 2 equals 3.
Note that while we have just chosen any values for our algebraic equations for the sake of these
examples, in practice your algebraic equation will be determined according to the relationships
between your variables, and by what you are wanting to find out.
A particular kind of algebraic equation that you may come across often is a linear equation, or first
degree equation, which is an equation that only has variables raised to the first power. For example a
minus 3 equals 5 is a linear equation, as opposed to a squared minus 3 is equal to 5, which has a
raised to the power of two, or second power, and hence is not a linear equation.
When a linear equation only contains only variable, as in the examples in the table, then we can
determine the value of the variable and hence solve the equation. How to do this will be covered in
this topic.
Note that if you need any revision on the basics of algebra, please refer to either or both of the What
is Algebra? and Expanding Brackets and Factorising topics before continuing with this topic.
Rearranging Linear Equations
The key to solving linear equations is to remember that you want to get the variable, by itself, on one
side of the equals sign; this way you will know what it is equal to!
For example if you are asked to solve the algebraic equation c + 2 = 5, you want to get c by itself on
one side of the equals sign.
While for a simple equation like this you may be able to ‘see’ what the variable is equal to without
performing calculations, this is often not possible for more complex examples. Hence it is important
to know how to solve it by rearranging the equation.
Rearranging an equation requires you to ‘get rid of’ constants that are around the variable; constants
that have been added to it or that it has been multiplied by, for example
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Linear Equations
The key to solving linear (or indeed any) equations is to remember that you want to get the variable,
by itself, on one side of the equals sign (typically the left hand side); this way you will know what it is
equal to! For example if you are asked to solve the algebraic equation c plus 2 equals 5, you want to
get c by itself on one side of the equals sign.
While for a simple equation like this you may already be able to ‘see’ what the variable is equal to
without performing any calculations, or to determine it by trial and error, this is often not possible for
more complex examples. Hence it is important to know how to solve it by rearranging the equation.
Rearranging an equation just requires you to ‘get rid of’ constants that are around the variable;
constants that have been added to it, like the 2 that has been added to c in our example, or that it
has been multiplied by, for example.
Undoing Operations
To do this, you need to undo whatever has been done with the constant by performing an ‘opposite’
operation. For example in an equation you might:
Remove a constant that
was…
Added
Subtracted
Multiplied by
Divided by
By…
Subtracting it
Adding it
Dividing by it
Multiplying by it
The important thing to remember here is that whichever operation or operations you perform on
one side of the equation, you must also do on the other side!
You might like to think of the equals sign as the pivot of an old-fashioned set of balancing scales, and
to remember this golden rule:
KEEP THE EQUATION IN BALANCE
To remove constants from around a variable, you need to undo whatever has been done with the
constant by performing an opposite operation.
For example, we might remove a constant that was added to the variable by subtracting it, remove a
constant that was subtracted from the variable by adding it, remove a constant that the variable was
multiplied by by dividing by it, or remove a constant that the variable was divided by by multiplying
by it.
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Linear Equations
The important thing to remember here is that whichever operation or operations you perform on
one side of the equation, you must also do on the other side! So you can’t add to, subtract from,
multiply or divide any constants or variables on one side of an equation, without also doing the same
thing on the other side of the equation.
To help you remember this you might like to think of the equals sign in an equation as the pivot of an
old-fashioned set of balancing scales. With this in mind, the golden rule is that the equation must
always be kept in balance.
Solving Linear Equations
In other words, make sure that whatever you do to one side of an equation you do simultaneously
(i.e., in the same step of working) to the other side.
This golden rule is demonstrated as follows:
On the left hand side (LHS) of
=, if you…
Add 2 (+2)
Subtract 5 (-5)
Multiply by 7 (x7)
Divide by 3 (÷3)
On the right hand side (RHS) of =,
you must…
Add 2 (+2)
Subtract 5 (-5)
Multiply by 7 (x7)
Divide by 3 (÷3)
So let’s go back to our equation c + 2 = 5. To solve it you would undo the addition of 2 by subtracting
2 from both sides of the equation, as follows:
c+2=5
∴c + 2 - 2 = 5 – 2 (undo addition by subtracting)
∴
c=3
(simplify)
Note that the ‘∴’ sign shown above is just shorthand for ‘therefore’.
What exactly do we mean by keeping the equation in balance? This just means that whatever you do
to one side of an equation you need to simultaneously, that is, in the same step of working, to the
other side as well.
This golden rule is demonstrated in the table shown. For example, if you add 2 to the left hand side of
an equation, you must also add 2 to the right hand side of the equation in the same step of working.
Similarly, if you subtract 5 from the left hand side of an equation you must also subtract 5 from the
right hand side, if you multiply the left hand side of an equation by 7 you must also multiply the right
hand side by 7, and if you divide the left hand side of an equation by 3 you must also divide the right
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Linear Equations
hand side by 3, and vice versa when performing these or other operations on the right hand side of
an equation.
Now that we have covered these key rules, let’s go back to our equation of c plus 2 equals 5. To solve
this equation by rearranging, you would undo the addition of 2 by subtracting 2 from both sides of
the equation. You should demonstrate this working by first writing out the original equation, c plus 2
equals 5, then writing out the equation again but showing subtracting 2 from both sides, and finally
performing the subtraction operations from the previous line of working, so that you are left with just
c on the left hand side of the equation, and 3 on the right hand side as your solution.
Note that the symbol with three dots shown with each line of working out is just shorthand notation
for therefore. Also note that the text in brackets is for explanation purposes only, and you are not
required to include such notes when providing your working. Once you get the hang of rearranging
and solving equations you may also be able to leave out some steps of working when writing up your
solution, but for the time being it is best to include everything so that any mistakes can be easily
identified.
Steps for Solving Linear Equations
Often multiple operations need to be undone in order to solve a linear equation. While this can
require a bit of thought for complex equations, a general starting point is to follow these steps:
1) If your equation has brackets in it, expand these first.
For example, if you wish to solve the linear equation 10d - 2 = 7(d + 1), your first step of working
would be:
10d - 2 = 7(d + 1)
∴
10d – 2 = 7d + 7
(expand brackets)
2) If your equation has like terms in it, group these together next (adding or subtracting them from
one side of the equation if required) and then simplify the equation.
For example, the next steps in solving the linear equation 10d - 2 = 7(d + 1) are:
∴10d – 2 – 7d = 7d + 7 – 7d (group like terms on LHS by subtracting)
∴
3d – 2 = 7
(simplify)
3) If your equation requires more than one constant to be removed from the variable, undo one
operation at a time in the following order:
a) Remove anything that has been added to or subtracted from the variable
b) Remove anything that the variable has been multiplied or divided by
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Linear Equations
For example, the next steps in solving the linear equation 10d - 2 = 7(d + 1) are:
∴ 3d - 2 + 2 = 7 + 2
(undo subtraction by adding)
∴
3d = 9
(simplify)
∴
3d/3 = 9/3
∴
d=3
(undo multiplication by dividing)
(simplify)
The first step in solving a linear equation is to expand any brackets in the equation first. For example,
if you wish to solve the linear equation 10d subtract 2 equals 7 times, in brackets, d plus 1, you would
first expand the brackets on the right hand side of the equation to give 10d subtract 2 equals 7d plus
7. Note that if you need more information about how to expand brackets, you should refer to the
Expanding Brackets and Factorising topic.
The second step in solving a linear equation is to group any like terms together on one side of the
equation, by adding or subtracting them from one side of the equation as required, and then
simplifying. For example, you would group the like terms in the equation 10d subtract 2 equals 7d
plus 7 together on the left hand side of the equation by subtracting 7d from both sides, to give 10d
subtract 2 subtract 7d equals 7d plus 7 subtract 7d. Simplifying this equation then means subtracting
7d from 10d on the left hand side, to give 3d, and subtracting 7d from 7d on the right hand side to
remove the variable d from that side of the equation, therefore giving 3d subtract 2 equals 7.
The final step in solving a linear equation is to consider what operations need to be removed from
the variable, and to undo them as appropriate. It is very important to note here though that if the
equation you are solving requires more than one constant to be removed from the variable you will
need to undo one operation at a time, first by removing anything that has been added to or
subtracted from the variable, and then by removing anything that the variable has been multiplied or
divided by. For example, the equation 3d subtract 2 equals 7 has a constant subtracted from and
multiplied by the variable. You need to undo the subtraction of 2 first by adding, so you add 2 to both
side of the equation to give 3d subtract 2 plus equals 7 plus 2, which simplifies to 3d equals 9 when
the addition operation is performed. Finally, you can undo the multiplication of 3 by dividing, so you
divide both side of the equation by 3 to give 3d divided by 3 equals 9 divided by 3, which simplifies to
d equals 3 when the division operation is performed. Hence 3 is the value of the variable d in your
equation.
Examples: Rearranging and Solving Linear Equations
1) c + 3 = 9
∴c + 3 – 3 = 9 – 3 (undo addition by subtracting)
∴
c=6
(simplify)
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Linear Equations
2) 3d – 2 = 7
∴3d - 2 + 2 = 7 + 2 (undo subtraction by adding)
∴
∴
3d = 9
(simplify)
3d/3 = 9/3 (undo multiplication by dividing)
d=3
∴
(simplify)
3) a/3 = 2
∴3(a/3) = (3)2 (undo division by multiplying)
∴
a=6
(simplify)
4) 3a + 4 = a + 6
∴ 3a + 4 – a = a + 6 – a (group like terms on LHS by subtracting)
∴
2a + 4 = 6
∴ 2a + 4 – 4 = 6 – 4
(simplify)
(undo addition by subtracting)
∴
2a = 2
(simplify)
∴
2a = 2
(undo multiplication by dividing)
2
∴
2
a=1
(simplify)
Let’s work through some more examples of rearranging and solving linear equations.
Example one requires us to solve c plus 3 equals 9. To do this we need to undo the addition of 3 to
the variable c by subtracting 3 from both sides, to give c plus 3 subtract 3 equals 9 subtract 3.
Simplifying the equation by performing the subtraction operation then gives c equals 6, which is the
solution to the equation.
Example two requires us to solve 3d subtract 2 equals 7. To do this we need to undo the subtraction
of 2 from the variable d first by adding 2 to both sides, to give 3d subtract 2 plus 2 equals 7 plus 2.
Simplifying the equation by performing the addition operation then gives 3d equals 9. We can then
undo the multiplication of the variable by 3 by dividing both sides by 3, to give 3d divided by 3 equals
9 divided by 3. Finally, we can simplify the equation by performing the division operation to give d
equals 3, which is the solution to our equation.
Example three requires us to solve a divided by 3 equals 2. To do this we need to undo the division of
the variable a by 3 by multiplying both sides of the equation by 3, to give 3 times a divided by 3
equals 3 times 2. Simplifying the equation by performing the multiplication operation then gives a
equals 6, which is the solution to the equation.
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Linear Equations
Example four requires us to solve 3a plus 4 equals a plus 6. To do this we need to first group the like
terms together on the left hand side of the equation, by subtracting a from both sides. This gives 3a
plus 4 subtract a equals a plus 6 subtract a, which simplifies to 2a plus 4 equals 6 when both sides of
the equation have the subtraction of like terms performed. Next we need to undo the addition of 4
by subtracting 4 from both sides, to give 2a plus 4 subtract 4 equals 6 subtract 4. Simplifying the
equation by performing the subtraction operation then gives 2a equals 2. We can then undo the
multiplication of the variable by 2 by dividing both sides by 2, to give 2a divided by 2 equals 2 divided
by 2. Finally, we can simplify the equation by performing the division operation to give a equals 1,
which is the solution to our equation.
Activity 1: Practice Questions
Click on the Activity 1 link in the right-hand part of this screen.
Now have a go at rearranging and solving linear equations on your own by working through some
practice questions.
End of Topic
Congratulations, you have completed this topic.
You should now have a better understanding of rearranging and solving linear equations.
Congratulations, you have completed this topic.
You should now have a better understanding of rearranging and solving linear equations.
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