A variable is a symbol, usually a letter, that represents a number. A variable can represent any number. In the algebraic expression 7 + x, x is a variable. Likewise, variables can appear in equations. In the algebraic equation 7×z = 42, s is a variable. If we replace s with the number 6, the equation will be true, since 7×6 = 42. We use variables in algebraic expressions when the quantity of something is unknown. For example, if I want to talk about "Peter's monthly salary plus $200," but I do not know his monthly salary, I might write s + 200, where s represents Peter's monthly salary. Or, if I want to say, "My book weighs three times as much as yours, plus 5 lbs. more," but I do not know how much your book weighs, I might write that my book weighs 3y + 5, where y represents the weight of your book. We also use variables to represent unknowns in algebraic equations. Consider the statement, "If Greg grows 2 inches, he will be 60 inches tall." This statement says that Greg's current height plus 2 inches is equal to 60 inches. We write h + 2 = 60, where h represents Greg's current height. Algebraic expressions and algebraic equations can contain more than one variable. For example, 2(h + w) is an algebraic expression that contains two variables, h and w.
Sometimes, we are given a known quantity for a variable. Look at the last slide and you’ll see that I might discover that Peter's monthly salary is $600, or that your book weighs 12 lbs by evaluating the expressions with certain numbers. To evaluate an algebraic expression, plug in the known quantity for the variable and evaluate the resulting expression. For example, given the known quantities described above, s + 200 = 600 + 200 = 800 and 3×w + 5 = 3×12 + 5 = 36 + 5 = 41. Every time you plug a number into a formula, like the formula for perimeter of a square, p = 4×s, you are evaluating an expression. What is the perimeter of a square with side length 1? 5? 2.5? p = 4×s = 4×1 = 4 p’ = 4×s = 4×5 = 20 p’’ = 4×s = 4×2.5 = 10
When we solve an algebraic equation, instead of plugging in a given number for the variable, we find a number that, when plugged in for the variable, would make the equation true. Such a number is called a solution to an equation. 58 is a solution to the equation h + 2 = 60, because 58 + 2 = 60. 46 is not a solution to h + 2 = 60, because 46 + 2 does not equal 60. Some equations have more than one solution. For example, 4 and -4 are both solutions to r2 = 16. Most of the equations we will deal with, however, have only one solution. The goal in solving an equation is to get the/a variable by itself on one side of the equation and a number/simplified expression on the other side. Generally, the variable will start on one side with operations being performed upon it. We must “eliminate” such operations by performing the inverse of each operation with the previous values of it in both sides of the equation, so that they remain equal. Operation multiplication division addition subtraction
Reverse division multiplication subtraction addition
You should be aware of the order of the operations for solving expressions. Solving an algebraic equation requires the reverse of such order. Addition or Subtraction (whichever one appears lastly); Multiplication or Division (whichever one appears lastly); Exponents; and lastly, Parenthesis.