# How to Solve an Equation

Learning how to solve an equation in various circumstances in Algebra follows on from the foundations of evaluating expressions and builds on it.

With equations there is now an equals sign ( = ) present, so we will start to see equations such as,

x \space {\text{–}} \space 4 = 3    and    3a = 12.

### How to Solve an Equation Approach:

To solve an equation in Algebra, we want to establish the value of the variable or variables that are present in the equation.
When learning how to solve an equation, we want to be able to get a variable by itself on one side of the equals sign.

We can do this by performing certain operations on the equation, but we must do these operations to both sides of the equals sign, so as to keep the overall equation value the same.

To understand this concept we can just look at a number.

12 = 12

Now adding say 3 to just one side messes things up.     12 + 3 = 12   ,   15 = 12

Clearly 15 does not equal 12.

But adding 3 to both sides, would be fine and keep everything right.

12 + 3 = 12 + 3   ,   15 = 15

We can see this process in action in some proper examples of how to solve equations below.

Examples

(1.1)

Solve   x + 2 = 8.

Solution

On the left with the variable we have a  2,  which we want to eliminate.

Subtracting  2  from both sides of the equation will do this.

x + 2 \space {\text{–}} \space 2 \space = \space 8 \space {\text{–}} \space 2

x = 6

An x value of 6 should work in the equation.

It’s good practice to check the value found by inputting it in the equation.

6 + 2  =  8   is true.

(1.2)

Solve   6a = 42.

Solution

Here the variable  a  has a multiplying coefficient of  6  that needs taken care of.

A division by  6  will undo the multiplication.

{\frac{6a}{6}} \space = \space {\frac{42}{6}}

a = 7

Checking like before to be sure, we have,   6 \times 7 \space = \space 42.

(1.3)

Solve   {\frac{x}{2}}  =  7.

Solution&nbsp &nbsp

Here we have a division on the left to undo, this can be done with multiplication.

{\frac{x}{2}} × 2  =  7 × 2

x  =  14

If we check by carrying out the original division, it is indeed the case that   {\frac{7}{2}}  =  14.

(1.4)

Solve   {\frac{4x}{3}}  =  12.

Solution

There is both a multiplication and a division here, so we’ll need to do both operations in order to get the variable on it’s own.
Order doesn’t matter, but it’ll possibly be easier to deal with the division first.

{\frac{4x}{3}} × 3  =  12 × 3

4x  =  36

Now we can deal with the x coefficient of 4.

{\frac{4x}{4}}  =  {\frac{36}{4}}

x \space = \space 9

Again doing a check.

{\frac{4(9)}{3}}  =  {\frac{36}{3}}  =  12

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