Solving Equations
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Solving Equations

Prentice Hall's Online Lesson Quiz   Enter Web Code: aga–0103

In this section we will review translating verbal expressions into algebraic expressions,  using properties of algebra, solving equations, and solving formulas for a specified variable.

ALGEBRAIC EXPRESSIONS:

Write an algebraic expression to represent each verbal expression below:

twice the sum of a number and 5

2(n + 5)
the product of the square of a number and 3.

3n2
seven decreased by four times a number

7 – 4n  

ALGEBRAIC PROPERTIES


PROPERTY


MODEL

REFLEXIVE PROPERTY OF EQUALITY

For any real number a, 
a  = a
SYMMETRIC PROPERTY OF EQUALITY

For all real numbers a and b, 
if a= b, then b= a
TRANSITIVE PROPERTY OF EQUALITY

For all real numbers a, b, and c, 
if a = b and b = c, then a = c

SUBSTITUTION PROPERTY OF EQUALITY

If a = b, then 
a may be replaced by b

ADDITION AND SUBTRACTION PROPERTIES OF EQUALITY

For any numbers a, b, and c, 
if a= b, then
a + c = b + c and
a – c = b – c
MULTIPLICATION AND DIVISION PROPERTIES OF EQUALITY

For any numbers a, b, and c, 
if a = b, then
a · c = b · c and,
 

Examples of some properties are shown below:

If x = 2y, then 3x= 6y Multiplication Property of Equality

If 5 + m = 3, then m = –2

 Subtraction Property of Equality
If w + (4 + 5) = 13, 
then w + 9 = 13		 Substitution Property of Equality
If 7 – 2 = 5 and 5 = 4 + 1,
then 7 – 2 = 4 + 1		 Transitive Property of Equality

An easy and helpful algorithm for solving equations in one variable:

  1. Is the equation a proportion?

         YES – Cross Multiply
    NO  –  Question 2

  2. Are there grouping symbols?

    YES – Distribute
    NO  –  Question 3

  3. Can the equation be simplified on either side of the equal sign?

    YES – Combine all like terms on either side of the equal sign
    NO  –  Question 4

  4. Are there variables on both sides of the equal sign?

    YES – Isolate all the variable terms to the side with the largest coefficient by using inverse operations
    NO  –  Question 5

  5. Are there numbers added or subtracted on the same side as the variable term?

    YES – Use the inverse operation to move these numbers to the other side, simplify and collect like terms
    NO  –  Question 6

  6. Are there numbers multiplied or divided on the variable?

    YES – If multiplied to the variable, then divide to get variable alone and equation is solved.  If divided on the variable, then multiply to get the variable alone and equation is solved.
    NO  –  Question 7

  7. Did the variable term disappear altogether?

    YES – Is the equation an identity (true statement) or is it a false statement, meaning equation has no solution.

SOLVING LITERAL EQUATIONS

Often students are confronted with formulas in science that need to be solved for a specific variable.  We can use algebraic properties to accomplish this task.  For instance, the distance formula states that distance is equal to the product of the rate of travel and time.  Namely, d = r · t.  However, we may know the distance traveled and the time it took to travel that distance and want to know the rate of travel.  This could be found by dividing both sides of the equation by time (t).  So the rate of travel is distance divided by time.  These common formulas can cause us trouble if we are not comfortable solving equations.  Let's look at some examples below:

EXAMPLE 1: Solve equation for b.

EXAMPLE 2:  Solve equation for a.