In algebra, letters are often used to stand for numbers that we want to find.  We call these letters variables or unknowns.  To use algebra, you must learn to translate words into algebraic expressions.  An algebraic expression may contain a variable as well as numbers and operations.  An expression may even contain more than one variable.  Any letter can be used as a variable, although some of the more common variables are a, b, c, x, y, and z.

Basic Algebraic Expressions:

Example

Expression

Notes

A number increased by four

x + 4 or

4 + x

Increased by means to add

Nine more than a number

r + 9 or

9 + r

More than means to add

Eight less than a number

y – 8

Less than means subtract

Eight decreased by a number

8 – y

Decreased by means to subtract (compare the order of this example to that of the last one)

Four times a number

4e

Times means to multiply (notice that there is no sign between the number and the unknown in multiplication expressions)

The product of ten and a number

10b

The product is the answer to multiplication

A number divided by 5

m/5

Notice that the divisor goes on the bottom

Five divided by a number

5/m

The unknown is the divisor here

One-fourth of a number

1/4m or m/4

Remember that 1/4 of a number is equal to that number over 4

The quotient of a number divided by 5

n/5

A quotient is the answer to division

A number times another number

xy

Two variables multiplied together

 


Practice writing algebraic expressions:
1. Five times a number 
2. The sum of a number and eight 
3. the product of a number and 4 
4. a number divided by 12  
5. a number decreased by 20  
6. ten times a number 
7. 14 subtracted from a number 
8. 45 decreased by a number 

Please note:  When you see the answer, the variable x will be used in all expressions, but you can use whatever variable you want.

9. Jack makes w dollars an hour.  He will get a $2-an-hour raise.  Write an expression for his new hourly raise.

10. Normally there are s students in John’s high school Equivalency class.  Because of bad weather, five students were absent one night.  Write an expression for the number of students who came to class that night.

11. Meat costs c per pound.  Write an expression for the cost of three pounds of meat.

12. Let i stand for Ellen’s gross income.  Of her income, 1/5 goes for taxes and social security.  Write an expression for the amount of Ellen’s income that is withheld for taxes and social security.

13. The town budget for summer programs is x.  Four programs share the money equally.  Write an expression for the amount each program gets.

14. Gail works 40 hours per week.  Which algebraic expression shows how much she earns, if z represents the amount she makes per hour?

15. Which expression shows the number of feet in a yard, if y represents the number of inches in a yard?

16. The sales tax in Bill’s state is 6%.  Which expression shows the amount of tax Bill has to pay on an item that costs m dollars?

17. The length of a rectangle is L.  The width is two inches less than the length.  Which expression shows the width of the rectangle?

Answers:  1. 5x;  2. x + 8 or 8 + x;  3. 4x;  4. x/12;  5. x – 20;  6. 10x;  7. x – 14;  8. 45 – x;  9. w + 2;  10. s – 5;  11. 3c;  12. 1/5i or i/5;  13. x/4;  14. 40z;  15. y/12;  16. 0.06m;  17. l – 2 (back to top)

Evaluating Algebraic Expressions
The value of an algebraic expression depends on what number is substituted for the variable.

For example, given the formula n + 5, if n = 6,

When we substitute 6 for n and get the answer 11.

When evaluating expressions, be sure to follow the same order of operations as directed above (you remember those, right???)

 

Let’s try a few together.
Example 1:  2a + 6 when a = 4

Step 1: Substitute 4 for the variable a, so your expressions becomes 2 * 4 + 6

Step 2:  Following the order of operations, we know that we have to do the multiplication first, 2 * 4 = 8

Step 3:  Then we can add the two number, 8 + 6 = 14

Our final answer is 14


Example 2:  3y2 + 3 when y = 5
Step 1:  Substitute 5 for y, so the expression becomes 3 * 52 + 3

Step 2:  Follow the order of operations, doing the exponent first, then the multiplication, and finally the addition.
            52 = 25
            3 * 25 = 75
            75 + 3 = 78

Our final answer is 78


Now try some on your own
1. –3(z + 5) when z = 4 
2. b – 5 when b= 10 
3. 2a + 6 when a = 4 

 

Find the value of the following expressions if a = 3, b = 9, and c = 18.

  4. a + b + c 
  5. abc
  6. 2a3 + b
  7. 5c – 2a2
  8. 6a – 8 
  9. 14a + 2b – c
  10. b2 – 2c + a 
  11. 4a / 2 + c
  12. a3 – (b + c) 
    1. Bart drove 60 mph for 2 hours.  Using the formula Distance = Rate * Time or d = r * t, find the distance Bart drove.
    2. Coleen worked twice as many hours this week as last.  She wrote the expression 2h to show the number of hours she worked this week.  If h = 25, how many hours did she work this week?
    3. Lisa’s bowling score for her first game (f) was 110, her second game (s) was 130, and her third game (t) was 135.  Using the formula (f + s + t)/3 = A, what is her average (A) for the three games?
    4. The cost per disk (r) to ABC Computer Company is $3.  The total cost (c) for the number of disks bought was $1,500.  The number of disks bought (n) was 500.  Which formula represents the situation?
    5. How much interest will you pay to borrow $4,250 (principal) for a year at a rate of 4%?  Use the formula Interest = Principal * Rate or I = P * R.

Answers: 1. –27;  2. 5;  3. 14;  4. 30;  5. 486;  6. 63;  7. 72;  8. 10;  9. 42;  10. 48;  11. 24;  12. 0;  13. 120 miles;  14. 50 hours;  15. 125;  16. c = nr;  17. $170 (back to top)

 

 

Resources:
GED Test 5: Mathematics. (2001). Contemporary Books, NTC/Contemporary Publishing Group: Lincolnwood, IL.
Math Skills Practice: Algebra. (1998). Cambridge Exercise Books, Cambridge Adult Education: Upper Saddle River, NJ.
Mitchell, Robert. (1988). Number Power 3: Algebra. Contemporary Books: Chicago, IL.