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# Forming and Solving Equations

### Forming and Solving Equations

Equations can help us to find the value of an unknown, by expressing the the equality of certain values. This is different to an identity.

Sometimes you will have a situation that can be best solved by forming an algebraic equation. An algebraic equation contains an equals sign, where the value of an **unknown letter** needs to be found.

For example:

$x+6=0$ is an equation, but $x+6$ is an expression.

This is different from an **identity** which is an equation where both sides are always the same value, no matter what value is substituted. An example of this would be $2x + 4 \equiv 2(x + 2)$.

Is $2x + 5 = 3 - 2x$ an equation or an identity?

Equations are a tool you can use to find an unknown value. The first step is to formulate an equation based on the information you have, and then to solve it using algebraic rules.

I think of a number, add 4 to it and then double the result. The answer is 36. What is the initial number?

Form an equation - $x$ is the unknown value

$2(x+4) = 36$ or$2x + 8 = 36$

Isolate the $x$term. First, divide both sides by 2

$\cancel2(x+4) = \cancel{36}$ so $x+4=18$

Subtract 4 from both sides

$x \space\cancel{+4} = 18-4$ so $x=14$

You've solved it!

$x=14$

Let's try another example! **I think of a number, divide it by 2, add 5 to it, then triple it. It is equal to 66. What is the original number?**

Form an equation

Isolate the $x$ term - $3\left(\frac{x}{2}+5\right)=66$

By isolating the $x$ term, we find $\frac{x}{2}=17$

We can find this by dividing both sides by 3, then subtracting 5 from both sides: $\cancel3\left(\dfrac{x}{2}\space \cancel{+5}\right)=\cancel{66} \space \space 22-5$. Therefore, $\dfrac{x}{2}=17$.

Solve $\frac{x}{2}=17$ to find $x$

$x=34$

We can multiply both sides by 2: $x=2 \times 17=34$. Nice! This is the final answer.

I think of a number, divide it by 2, add 5 to it, then triple it. It is equal to 66. What is the equation that represents this?

Solve $3(\frac{x}{2}+5)= 66$