Why x and y are Simplifying Expressions?
You're doing algebra now, and you know you're going to see x's and y's. But before we work with x's and y's, we'll explore why we use them.
Let's look at this problem. Tony bought two items at a store. He checks the sales receipt and finds that the cost of one of the items is blurred. He sees he spent $12 in total, and one item costs $7. How much does the other item cost?
He thinks: 7 + x= 12
He tries different numbers. He rejects the numbers that don't work until he eventually finds one that does.
7 + 1 = 8 ≠12
7 + 2 = 9 ≠12
...
7 + 5 = 12
Trying different numbers until you find one that works is one way to solve this problem.
However, what if the total were $111.89
and one item cost $61.42: 61.42 +x = 111.89
or there were more than two items, for example: 4.99 + 9.53 + x= 18.47
or there were 2 of the same item, for example: 6.69 + 2 = 12.89
In these cases, and in most real-life situations, it would be impractical to try to work through a long string of possibilities until the right one is found. Algebra provides a quick, systematic way to find a solution to problems like these (and many others).
In the equation 7 + = 12, the serves as a placeholder for a number. Its value is variable. The number 5 makes the equation true; other numbers make it false.
The equation 7 + x = 12 says exactly the same thing. Here, x is the placeholder for the different values that can be used in the equation. When x = 5, the equation is true. We could have used other symbols as well: 7 + y = 12, 7 + ? = 12, etc.
For the purpose of holding a place for a number in an equation, letters do seem to work better than other symbols. By convention, we use x, y, z, most often, but any letter can serve as a placeholder or variable. In algebra, you will be asked to solve an equation, which means finding the value or values of a variable that make an equation true.
The equation y + z = 9 is a another type of algebraic statement. When two or more letters appear, they each serve as a placeholder for a different number. Notice that there are many values for y and for z that make this equation true.
For example: y = 0 and z = 9, y = 1, z = 8, etc.
Different letters in an algebraic expression can take on different values. The same letter has the same value no matter how many times it appears in an algebraic expression.
You might think of letters, or variables, as numbers in disguise. Whatever is true for numbers is true for letters (variables). Use all the Operations of Arithmetic, Adding, Subtracting, etc., for letters as you would for numbers. So, the following would be true.
x + x = 2x, which we write as x + x = 2x
7y - 3y = 4y
4z/4 =z
3x + 4y - 2x = 4y + x