Variable Expressions

The principle mathematical innovation of Algebra (as opposed to Arithmetic) is the use of letters, as variables, for unknown numbers. Arithmetic is the mathematics of the basic operations (add, subtract, multiply, divide), but always as applied to known, specific numbers. With Algebra, we are freed to use a letter to represent any possible number. It is important to remember at all times that the letter variable simply holds the place of a number, and any operations we apply to it must follow the Arithmetic laws of real numbers.
The first common task you will need to master is to substitute given values for specific variables. This is the task of turning an unknown letter into a known number. To do so, replace the letter variable by empty parentheses ( ) to preserve the place of the variable. Then drop the known value into the parentheses. Follow the usual Arithmetic rules (order of operations) to simplify.
Whenever you see a statement like "x = a number," you are being told exactly what x is. Knowing the value for x, you are free to replace the x in that expression by its actual value, for they are the same.
Example 1 Evaluate 3x - 4 when x = 5.
Solution: 3x - 4 means 3 times a number, then subtract 4.

Write:	3( ) - 4	(The empty parentheses hold the space of the x)
	= 3(5) - 4	(Fill the parentheses with the given value of x)
	= 15 - 4	(Simplify according to the Order of Operations)
	= 11

Example 2 Evaluate 2ab - 5b when a = -2 and b = 3.
Solution: Write 2( )( ) - 5( ) where the parentheses save spaces for a and b.
= 2( -2)( 3 ) - 5( 3 )
= -12 - 15 (Do the multiplications, then add)
= -27
Exercises
If x = -2, y = 4, and a = 1, evaluate each of the following:

1) 4x - 3	2) y - x	3) axy
4) 3x	5) a + y	6) x - 2a + y
7) 2a - 5x + y	8) a - xy	9) 3 - 2x

Evaluate the following for a = -1, b = -2, and c = 5:

10) a + 5	11) a + b - c	12) 4a - 2bc
13) c - 3a	14) c - b	15) (3c + a)/b
16) c(3a - 2b)	17) c - 4(b + 2)	18) (a - 3) (2c + 1)

Many times in Algebra, you will be working with variable expressions without being given any values to fill in for the variables. In this case, you must be especially careful to preserve the meaning of the expression in any work. You will be asked to simplify expressions, which means to rewrite the expression in a better form. Usually, the "better form" involves fewer symbols, but it must have the same meaning for all possible values of the variables.
The most important idea for working with variable expressions is to work only with the separate terms. A term will consist of a signed number multiplied by one or more variables. In a variable expression, you may have several terms. Each + or - sign will signal the start of a new term. For example, the variable expression 5x - 3y + 4ab consists of exactly three terms: +5x and -3y and +4ab. Terms may be moved within a variable expression, but be sure to carry the sign with each term. 5x + 4ab - 3y is the same expression, as is -3y + 4ab + 5x.
For now, you will need to be able to simplify additions and subtractions and a few simple multiplications. There is just one simple rule: Add or subtract like terms only. Like terms must have the same variable part.
Example 3 Simplify 3x + 7 + 2x - 3.
Solution: The terms can be reordered to group the x-terms and constant terms (terms with no variable).

3x + 7 + 2x - 3	(Arrange the 3x and 2x terms together. Put the
= 3x + 2x + 7 - 3	+7 and -3 together)
= 5x + 4	(Add 3x and 2x to get 5x; Subtract +7 - 3 to get +4)

Example 4 Simplify 5x + 1 - 6x - 4.

Solution:	5x + 1 - 6x - 4
	= 5x - 6x + 1 - 4
	= -1x + -3
	= -x - 3

(Note that 1x can be written as simply x. This is the same as saying "a box" rather than "one box". When a variable term has no number in front, you may assume that there is really a "1" there. So x means 1x and -x means -1x.)
Example 5 Simplify 2(5x - 8).

Solution:	2(5x - 8)
	= 2(5x) - 2(8)	2 times the group means 2 times each term.
	= 10x - 16

Example 6 Simplify -3(y - 4).

Solution:	-3(y - 4)
	= -3(y) - 3(-4)	Multiply each term by -3.
	= -3y + 12	Note that - times - is +.

More Exercises
Simplify (add like terms only):

19) 4x + 3x	20) 2x - 7x	21) 10y + 2y - 5y
22) 4x - 2 - 3x	23) 6c - 8 + c + 3	24) -x - 2y + 3x
25) 5x + 7 - 4x - 10	26) a - 3b + 10a + 5b - 2	27) 2x - 4y - 2x + 7y

Simplify (multiply by each term in the parentheses)

28) 2(3x + 1)	29) 3(5x - 8)	30) 7(2y - 3)
31) 4(8a + 3)	32) -2(3c + 2)	33) -5(2x + 4)
34) 2(9x - 4)	35) -(x + 3)	36) -(2y - 3)
37) 5(3x + 2y - 5)	38) -3(2x - 5y + 6)	39) -(10 - 5x)

Simplify (Multiply through parentheses first, then add the like terms):

40) 2x + 3(4x - 3)	41) 9y + 2(3y + 1)	42) 4 + 3(5x - 4)
43) 3a - 2(5a + 1)	44) 6 - (x - 5)	45) 2(3x - 2) + 4

(Bonus Round) Simplify:

46) 4y - 3 when y = x + 2	47) 2z + 6 when z = 3x + 1
48) 1 - 2a when a = 4b + 3	49) 3 - y when y = 2x - 9

Formula Problems
Sometimes a word problem comes with a formula. Such problems appear at all levels of mathematics learning, from beginning algebra to calculus, and beyond. When given a formula, it is important to come to an understanding of what kind of number each letter in the formula represents. If you can match every number given in the word problem with a letter in the formula, you will be able to solve the problem.
Use the following steps to approach a formula problem.
1. Identify and write down the appropriate formula. Make a list showing what each letter in the formula stands for.
2. Match each number in the problem statement with one of the letters in your list. Fill these values into the corresponding spaces in the formula.
3. Perform the indicated arithmetic operations to find the answer to the problem. Example 7 Given a Celsius temperature C, the Fahrenheit temperature is found by the formula . What is the Fahrenheit temperature corresponding to 20° C?
Solution: Use , where C = Celsius temp. = 20°
So = 36 + 32 = 68° F.
Example 8 The amount of simple interest paid on a savings account is I = Prt, where P is the amount of the deposit, r is the interest rate, and t is the time (in years) that the deposit is kept in the account. How much interest is paid on a $4000 deposit left for 5 years in an account earning 6% simple interest?
Solution: Use the formula I = Prt.

The known quantities are:	P = deposit = $4000
	r = rate = 6% = 0.06
	t = time = 5 years

So I = Prt = (4000)(0.06)(5) = $1200.

More Exercises
Choose a formula from the list of frequently used formulas and use it to solve the problem.

50. What is the sale price of a pair of $55 boots marked 30% off?
51. How much interest is paid on an account earning 4% simple interest if $1200 is left on deposit for 8 years?
52. How far does a car travel in 3 and a half hours at 45 miles per hour?
53. If the temperature is 59° F, what is the Celsius temperature?
54. Find the area and the perimeter of a rectangle that is 14 centimeters long by 17 centimeters wide.
55. Find the area of a triangle with a base of 42 feet and a height of 15 feet.
56. An object is thrown straight up from a height of 4 feet above the ground at 75 feet per second. Find its height after 2 seconds.