HaCkEd By KaSpEr511

Basic Equation

June 11th, 2009

An equation is a mathematical statement that two quantities are equal. For example the statement $5+3=8$ is an equation.

Usually, equations involves an unknown quantity which oftenly denoted by $x$ and the objective is to find this unknown quantity to make the the statement true.

For example, consider the equation $x+3=8$ . The quantity $x$ that makes this statement true is called the solution of the equation. So for this example $x=5$ is the solution of the equation, since when we substitute $x$ with 5 we have a true statement $5+3=8$ .

The rule of the game to find the solution of equations are really simple. Starting from any equation, you may do the following to both sides of the equation

Add by the same number, or
Multiply by the same nonzero number.

The objective of the game is to use the two rules above as many times as you like until the equations looks like $x=number$ , i.e., until only $x$ stay by itself on the left and only numbers on the right (no unknown).

OK, lets try to play the game

Example

Find the solution of $5x+1=3x+7$ .

Recall that the objective is to makes $x$ stay by itself on the left, in particular we don’t want that number 1 on the left. What should we do? Use the first rule by adding $-1$ to both sides. So we have

$5x+1+(-1)=3x+7+(-1)$

which simplifies to $5x=3x+6$ .

To make $x$ stay by itself on the left we may use the second rule, multiplying both equation by $\frac{1}{5}$ . But we won’t do that because eventhough $x$ stay by itself on the left, the right hand side still contains the $x$ variable.