Mathematical Journey/Solving Equations Part 1/Resources

Let's first review some basic math principles.

For example:

 * $$2 + (-2) = 2-2=0$$
 * $$(-2) + 2 = 0$$
 * $$0=(-10)+10$$
 * $$0=64+(-64)$$

For example:

 * $$6+0 = 6$$
 * $$0+12 = 12$$
 * $$x+ 0 = x$$
 * $$0 + y = y$$

Principle 3: You can only combine like terms.

 * numbers are added to numbers, and
 * variables are added to variables.

For example:

 * $$5 + x + 7 = 12 + x$$, which may also be written $$x+12$$
 * $$ x + 5 - 5 = x + 0 = x$$
 * $$6 + x - 6 = 0 + x = x$$

Solving an equation
A variable is the unknown in an equation represented by a letter.

Solving an equation is like unwrapping a package, we unwrap the variable. In other words, we undo what was done to the variable. So, if 6 were added to the variable, we subtract 6 to get to the variable and find out what number it represents. Similarly, if 2 were subtracted from the variable, we would add 2 to unwrap the variable. The thing is, you must remember that if you perform an operation to one side of an equation, you must do the same thing to the other side.

Example 1: $$x + 5 = 85$$
Look on the side of the equation containing the variable. Since 5 was added to the variable x, we subtract 5 from both sides to get the x alone on one side of the equal sign. Then we can say that x= such and such. Here's how we solve the equation:

$$\begin{align} x + 5 & = 85 \\ x + 5 -5 & = 85 - 5 \qquad \text{subtract 5}\\ x + 0 & = 80 \qquad \text{combine -5 and 5, and 85 and -5} \\ x & = 80 \\ \end{align}$$

Example 2: $$x -4 = 12$$
The variable is on the left side of the equation. Since 4 is subtracted from the variable, we will undo that by adding 4 to both sides. Then we can say that x = 16.

$$\begin{align} x - 4 & = 12 \\ x -4 + 4 & = 12 + 4 \\ x + 0 & = 16 \qquad \text{typically, we skip this step} \\ x & = 16 \\ \end{align}$$

Example 3: $$24 = x + 14$$
In this example the variable is on the right side of the equation.

Since 14 was added to the variable, we will subtract 14 from both sides of the equation.

$$\begin{align} 24 & = x + 14 \\ 24 -14 & = x + 14 - 14 \\ 10 & = x \qquad \text{or} \ x = 10 \\ \end{align}$$

So, basically you are combining positives and negatives of a particular number to get them to cancel out to get the variable alone on one side of the equal sign.

Example 4: $$-15 + x = 34$$
The variable is on the left side of the equation with a -15 so you will add 15 to both sides.

$$\begin{align} -15 + x & = 34 \\ -15 + 15 + x & = 34 + 15 \\ x & = 49 \\ \end{align}$$

Example 5: $$55 = -10 + x$$
The variable is on the right side of the equation, along with a -10.

$$\begin{align} 55 & = -10 + x \\ 55 + 10 & = -10 + 10 + x \\ 65 & = x \qquad \text{or} \ x = 65\\ \end{align}$$