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Rearranging and solving linear equations


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Algebra

Rearranging and Solving Linear Equations

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This topic will cover how to:

  • 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:

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.

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.

5 + d5 + d = 10
3n + 13n + 1 = 4
6 - f6 - f = 5
a/2a/2 = 3

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.

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.

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:

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

AddedSubstracting it
SubtractedAdding it
Multiplied byDividing by it
Divided byMultiplying by it

KEEP THE EQUATION 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:

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

Add 2 (+2) Add 2 (+2)
Subtract 5 (-5) Subtract 5 (-5)
Multiply 7 (x7) Multiply 7 (x7)
Divide by 3 (/3) Divide by 3 (/3)
c + 2 - 2 = 5 - 2 (undo addition by subtracting)
c = 3 (simplify)

Note that the '' sign shown above is just shorthand for 'therefore'.

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
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 (undo multiplication by dividing)
d = 3 (simplify)

Examples: Rearranging and Solving Linear Equations

1)c + 3 = 9

2)3d - 2 = 7

3)a/3 = 2

4)3a + 4 = a + 6

c + 3 - 3 = 9 - 3 (undo the addition by subtracting)
c = 6 (simplify)
3d - 2 + 2 = 7 + 2 (undo the subtraction by adding)
3d = 9(simplify)
3d/3 = 9/3 (undo the multiplication by dividing)
d = 3(simplify)
3(a/3) = (3)2 (undo the division by multiplying)
a = 6 (simplify)
3a + 4 - a = a + 6 - a (group like terms on LHS by subtracting)
2a + 4 = 6(simplify)
2a + 4 - 4 = 6 - 4 (undo addition by subtracting)
2a = 2(simplify)
2a/2 = 2/2(undo multiplication by dividing)
a = 1(simplify)

Activity 1: Practice Questions

Now have a go at rearranging and solving linear equations on your own by working through some practice questions by clicking on the Activity 1 link in the right-hand part of this screen.

End of Topic

Congratulations, you have completed this topic.

You should now have a better understanding of rearranging and solving linear equations.

 


PDF Version PDF

Follow the link to view the ‘Rearranging and Solving Linear Equations’ pdf version.

 

 

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