Working with equations is an important step in laying the proper foundations for algebra. Look at this equation: 2 + 6 = . This equation is an open mathematical sentence. The box represents an unknown value. The intriguing idea is that the box can be replaced by another symbol such as ?. Now the equation becomes 2 + 6 = . The value for the box in the first equation is 8, as is the value of the triangle in the second equation. You could even alter the equation to 2 + 6 = ___. The box, triangle, and blank are variables. More commonly, a letter such as x is used as a variable. The variable can be used in the
middle of an equation, too:
Both sides of the balance must equal the same thing at all times for the pans to
balance. Notice the scales above are balanced because each pan has a five on it.
Now look at the next figure:
This scale is balanced too, because the total on each side is 5. This balance shows the equation 4 + 1 = 5.
3 + x = 8, for example.
A common misconception is to treat the equals sign as meaning "write the answer now." In an equation, the equals sign indicates that the quantities on each side must be equal. You can think of an equation as a pan balance, with the equals sign as the balance point.
It is important to understand that we want to keep the pans balanced. If we add two to the left pan above, the left side would go down while the right side would go up since 4 + 1 + 2 (which equals 7) is greater than or heavier than five.
But if we add two to both pans, the scales would remain balanced since both sides would yield a sum of 7.
The goal is to balance both sides of the equation. We want to have equivalent values of the left and right side of the equations. Think of the center bar as the equal sign. A key rule for working with equations is that you can add the same number to each side or multiply each side by the same number and the sides will stay equal. You must do exactly the same operation to each side, though, or the equation will become unbalanced.
Adding and Multiplying by Equals