Arithmetic versus Algebra
Arithmetic is our daily companion in life. We use it everyday. Such that, this topic is not to teach you something new, but only to extend your skill to different applications of the things you already knew.
If you consider arithmetic as a “hand tool” for mathematical work, algebra is the “power tool.” A handsaw is very useful for many small jobs of sawing, but when it comes to great sawing jobs like cutting a tree, for example, the power saw makes things better. What you can finish in one-half day with a handsaw, you can do it in an hour with the power saw. Like a handsaw, arithmetic is a good tool for solving simple problems. However, more involved and more advanced problems require a more powerful tool, ALGEBRA.
Letters in algebra
In algebra a letter stands for a number whose value is not known. For example, you may say “I have x dollars in my wallet.” You did not count your money, but you know you have some. The letter x stands for the unknown number. If you look in your wallet and find that you have two $10 bills and five $1 bills, you will know that you have $25. The value of x is $25. How would you write the rule for determining the value of x ? You can write x = 2 x 10 + 5 This expression is the formula that determines x in this case. When you perform first the multiplication and then the addition at the right side of the equals sign, you have “solved” the formula and found that x = 25.
Remember this rule about the sequence of operation: always multiply or divide before you add or subtract. Of course, not all formulas are that simple, but they all represent a mathematical rule for determining an unknown value. The expression x= 2 x 10 + 5 can also be called an equation because it indicates that the value (or the amount) on one side of the equals sign is equal to the value on the other side of the equal sign. “Solving” the equation means finding the value of x = 25.
What is an equation?
The easiest way to get acquainted with equations is through an example. Let’s assume that there are seven men and five women in the room. How many people are there in the room?
What is not known in this problem? The total number of people in the room. Let’s use the symbol x for this unknown number or quantity. What is the clue for finding the quantity represented by x ? Obviously, the quantity can be found by adding 7 and 5. How can this statement be expressed mathematically? The statement can be written x = 7 + 5. This mathematical expression is the equation for solving the given problem.
An equation is a mathematical statement that something is equal to something else. The equals sign (=) is the center of the equation. To the left of the equals sign in our example is the unknown quantity x, and to the right of the equals sign is the sum 7 + 5. If the two sides must be equal to each other, what should be the numerical value of x ? Add 7 and 5 and write the sum on the right side of the equation: x = 12.
This statement is the solution of the equation. It still has the form of an equation—the left side is equal to the right side—but now you can read directly: x equals 12. The unknown has become known and the equation has been solved. In this problem, the symbol x stands for 12 because 12 = 7 + 5.
To solve the equation it was necessary to use simple addition. You already knew how to add 7 and 5. The new thing in solving the equation was the use of x as a symbol for the number to be found. An equation expresses mathematically the relationship between known and unknown quantities. It states the equality of expressions on both sides of the equals sign. Assuming the equality to be true, the unknown quantity must be found by calculation.
Of course, not all equations are as simple as our example, but all equations are based on the same principle.