Introduction to equations

An equation is a mathematical statement that two expressions are equal. It is important to understand the significance of the equal sign in an equation. The equal sign tells us that the left side of the equation is equal to the right side of the equation. And, no matter what, that must always be true. Therefore, if you make a change to one side of the equal sign, you must make the same change to the other side. For example, in the equation 6 + 5 = 11, if I add 2 to the left side, I must also add 2 to the right side for the = to be true:

$$\begin{align} 6 + 5 & = 11\\ 6 + 5 + 2 & =11 + 2\\ 13 & =13 \\ \end{align}$$

Picture a ruler balancing on my finger. The left side equals the right side when the ruler is in balance. However, if I put an eraser on one side, the ruler is out of balance. To put it back in equilibrium, I must also put an eraser of equal size on the other side. The balanced ruler is like an equation where my finger is the equal sign.

In solving equations, it will become necessary to perform calculations (add, subtract, multiply, or divide) on the equation. Just remember, what you do to one side of the equation, you must do to the other side.

Let's subtract 2 from the equation 6 + 5 = 11 so we get

$$\begin{align} 6 + 5 & = 11\\ 6 + 5 -2 & =11 - 2\\ 9 & =9 \\ \end{align}$$

Now, multiplying both sides by 2

$$\begin{align} 6 + 5 & =11 \\ 2(6 + 5) & = 11*2 \qquad \text{multiply both sides by 2}\\ 2(11) & = 11*2 \qquad \text{remember PEMDAS--Please Excuse My Dear Aunt Sally; do what is inside the parentheses first}\\ 2 * 11 & =11* 2 \\ 22 & = 22\\ \end{align}$$

In the equation, $$2*3 = 6$$, if we divide one side by 2, we must divide the other side by 2 also.

$$\begin{align} 2*3 & =6 \\ \frac{2*3}{2} & = \frac{6}{2} \qquad \text{divide both sides by 2}\\ 3 & =3 \\ \end{align}$$