Introduction

A basic understanding of algebra is required in most introductory economics courses. In the course of your studies, you will need to perform fundamental operations including how to write a verbal statement as an algebraic equation, how to express averages, how to handle non-parenthetic and parenthetic expressions, how to use signed numbers correctly, and how to solve for an unknown variable in algebraic equations. Reviewing and understanding the ten basic algebraic rules that follow will help you to be better prepared to tackle any mathematical situations you encounter. The self-test comprises twenty-five questions that will test your understanding of these rules and can help you identify areas of difficulty. Their complexity increases, so allow a little more time for the final ones. Correct answers are provided along with tutorial-type solutions.

Basic Algebraic Rules

1. The negative of a negative value is positive.

Example: -(-b) = b

(You can put in a number, say 3, for "b," if you prefer to use numbers rather than variables.)

 What is really going on is that the negative sign in front of (-b) is really -1. So another way to think of the rule is:

2. Multiplication of a negative value by a negative value returns a positive value.

Example: (-a) * (-b) = a * b.

(Here again, if you prefer, you can put in numbers for a and b. If a = 1 and b = 3, then (-1) * -3 = 3.)

3. Multiplication of a negative value by a positive value returns a negative value.

Example: (-a) * (b)=-a * b.

(Here again, if you prefer, you can put in numbers for a and b. If a = 4 and b = 3, then (-4) * 3 = -12.)

4. When moving variables in an equation from one side of the "=" sign to the other, the sign of the variable will change. If the variable was positive, it will now have a negative sign in front of it. If the variable was negative, it will now have a positive sign in front of it.

Example: Y = a - b * X

This can also be written as:
Y - a = b * X

In the example above, "a" was eliminated from the right-hand side of the equation by effectively subtracting it from each side of the equation (as long as you do the same thing to both sides of the equation, its meaning is not altered). That is:

Y - a = a + b - X - a.

Since a - a = 0, the "a" on the right hand side of the equation is eliminated. Other ways in which the equation Y = a - b * X can be written are:

Y + b * X = a
Y - a + b * X = 0.

5. A variable multiplied by its reciprocal equals 1.

Example: J * (1/J) = 1.

6. If two variables are multiplied by each other, separating them requires division.

Example: Y * Z = H

This can also be written as:

Y = H/Z.

In the example above, Z is effectively eliminated from the left hand side of the equation by dividing the left and right-hand side of the equation through by Z. (As long as you do the same thing to both sides of the equation, its meaning is not altered.) Since Z/Z = 1, Z is eliminated from the left-hand side of the equation.

7. Division by a variable is equivalent to multiplying by the reciprocal of the variable.

In the example to rule 4 above, Z was eliminated on the left-hand side of the equation by dividing through each side of the equation by Z. You can also think of this as multiplying each side of the equation by 1/Z. (As long as you do the same thing to both sides of the equation, its meaning is not altered.) That is:

1/Z * Y * Z = H * (1/Z)

Since 1/Z * Z = 1, the equation becomes Y = H/Z.

8. When a set of variables is multiplied (or divided) by a common variable, the common variable can be factored out.

Example: Z - d * Z + h * Z

This can also be written as: Z * (1 - d + h)
Where Z is the common variable.

Example: k/q - d/q + h/q

This can also be written as: (1/q) * (k - d + h) Where 1/q is the common variable.

9. When given two equations with a common variable, substitutions can be made.

Example: Y = a + b * X and X = z - v * U

You may substitute for X from the second equation into the first equation to get:

Y = a + b * (z - v * U)

Y = a + b * z - b * v * U

Factoring the "b" out of the two last terms on the right-hand side of the equation yields:
Y = a + b * (z - v * U)

10. Variables that appear on both sides an equation should be moved to the same side where factoring can be done.

Example: Y = a + b * z - b * v * Y

The common variable is Y. By moving the Y term on the right-hand side (along with any variables that it is multiplied or divided by) to the left-hand side (see rule #2 above), the equation becomes:

Y + b * v * Y = a + b * z

Factoring out the "Y" on the left-hand side (see rule #8 above) gives:

Y * (1 + b * v) = a + b * z.

The term (1 +b * v) can be eliminated from the left-hand side of the equation by dividing through both sides by (1 +b * v). (See rule #6 above). This yields a "solution" for Y as:

Y = [a + b * z]/[1 + b * v]

Self-Test

1. What is 4 times x decreased by one-fifth of y?
A. 4 + x - 1/5y
B. 4x - 5y
C. 4x - 1/5y
D. 1/5y - 4x

2. What is half the sum of x and twice y, expressed algebraically?
A. x + 2y / 2
B. 2x + y / 2
C. x + y / 2
D. x + 2y/0.5

3. How would you express the average of x, y, and 80, in algebra?
A. x + y + 80
B. x + y + 80 / x + y
C. x + y + 80 / 2
D. x + y + 80 / 3

4. Using an algebraic expression, what is one-third of the product of 7 and x expanded by 100?
A. (7/3 + x) * 100
B. (7/3 + x) + 100
C. (1/3 * 7) + 100
D. 7x/3 + 100

5. In algebra, what is the quotient of x and 100, minus 4 times the sum of x and 100?
A. x/100 - 4(x+100)
B. 100x - 4(x + 100)
C. x+100 = 4(x + 100)
D. x+100 = 4+x+100

6. Using algebra, how would you express a price that is 50 cents cheaper than another of x dollars?
A. 100x - 0.5
B. 50x - 100
C. ½ - 100x
D. 100x - 50

7. Using algebra, how would you express a price which is 25 cents more than another of y dollars?
A. y + 25
B. 100y + 25
C. 25 + y
D. 25y

8. What is the product of 50 and the sum of x and 10?
A. 50(x + 10)
B. x + 10 + 50
C. x + 10/50
D. 50(10x)

9. How would you express the average of x and 50?
A. 2(x + 50)
B. x + 50/2
C. 50 + 2/x
D. 2/50x

10. What is the value of x + 10y - z/5, when x=10, y=5, and z=100?
A. 10
B. 20
C. -5
D. 40

11. What is the product of 2(4 + 3)?
A. 9
B. 11
C. 14
D. 10

12. What is the value of 3(x + 2b)-8, when x=12 and b=2?
A. 4
B. 8
C. 32
D. 40

13. What is the numerical coefficient in the algebraic term, 5y / 7x?
A. 7/5
B. 5/7
C. 5
D. 7

14. What is another way to state this term: - (-5x)?
A. 5x
B. -5x
C. 5-x
D. 5/x

15. What is the product of: -x * -x?
A. -x
B. 2x
C. 2-x
D. x2

16. What is the product of: -50 * 2?
A. 100
B. -50/2
C. -100
D. 2/-50

17. What is the product of -50 * -100?
A. 5000
B. -5000
C. ½
D. 2

18. Solve for x in the following equation: x + 5=50.
A. 55
B. 50/5
C. 5/50
D. 45

19. Solve for x in the following equation x - 5=50.
A. 45
B. 55
C. -10
D. 10

20. How can we properly separate the two variables, x and y, from the following equation, x*y = z?
A. x=z/y
B. xyx = 1
C. xyx = 0
D. x = zy

21.What is the product of this variable when it is multiplied by its reciprocal? y * 1/y.
A. y
B. 1
C. 2y
D. y/1

22. What is the result of dividing x by 0.5?
A. 0.5x
B. x+0.52
C. 2x
D. 2

23. Solve for y in the following equation, y * 1/z * z = h * 1/z (see question #21 for a hint!)
A. y = z/h
B. y = 2zh/x
C. y = h*1/z
D. y = h/z

24. How can this product be expressed more simply? Y - a * Y +b * Y?
A. Y*(1 - a + b)
B. Y + (1 - a + b)
C. Y - (1 - a + b)
D. Y/(1 - a + b)

25. Simplify y to a fraction from the expression, y = a + b * z -b * v * y.
A. y = a + b*z/-b*v
B. y = a + b*z/1 + b*v
C. y = a + b/ 1 - b*v
D. Y = a + b/ 1 + b*v