3 Equations

3.1 Introduction to Equations

An equation is a statement that asserts two quantities are equal. The left hand side is equal to the right hand side. $ x = 4 $, $ p+3 = -11 $, $ -2z = 20 $, $E=mc^2$ are examples of equations.

When solving an equation the aim is to get a single variable on the left hand side equal to an expression on the right. Addition, subtraction, multiplication and division operations can be used to solve an equation as long as they are applied equally to both sides of the equation. This is what keeps both sides equal.

Example 3.1: Rearranging equations

Rearrange the following equation to get $y$ equal to a function of $x$.
$2x+3y-4$$=$$6x-y+2$
First, add $4$ to both sides
$2x+3y-4+4$$=$$6x-y+2+4$
$2x+3y$$=$$6x-y+6$
Next, add $y$ to both sides
$2x+3y+y$$=$$6x-y+y+6$
$2x+4y$$=$$6x+6$
Now, subtract $2x$ from both sides
$2x-2x+4y$$=$$6x-2x+6$
$4y$$=$$4x+6$
Divide both sides by $4$
$4y/4$$=$$(4x+6)/4$
$y$$=$$(4x+6)/4$
Finally, notice all the terms on the right are even so we can simplify
$y$$=$$x+3/2$

As long as the operations are applied equally to both sides of the equation you can do anything to get the equation in a useful form.

3.2 Equations with Radicals

A radical is a root that contains a variable like $\sqrt{x}$ or $\sqrt[3]{y}$. A root that contains only constants is just a constant.

To solve an equation with radicals we need to get the variable of interest on the left hand side of the equals sign and an expression that does not contain the variable of interest on the right. Then, and only then, we can square both sides of the equation to get an expression for our variable of interest. As an example, make $y$ the subject of $r=\sqrt{x^2+y^2}$

Example 3.2

Make $y$ the subject of $r=\sqrt{x^2+y^2}$

First we square both sides of the equation.

so $r^2=x^2+y^2$

Then we subtract $x^2$ from both sides.

$r^2-x^2=y^2$

Swapping the left and right expressions

$y^2=r^2-x^2$

Finally we take the square root of both sides

$y=\sqrt{r^2-x^2}$