Lecture Notes for Section R.7, Equations

Up to now we have manipulated algebraic expressions, now we will use those skills to solve algebraic equations.

Definitions

1. An equation is a statement of equality. It works like a balance beam in that if you do the same thing to both sides it is still in balance. This allows us to isolate variables (i.e. solve for them).
2. A number of an expression that makes an equation true when substituted in for the variable is a solution.
3. Root is another word for solution. Root is used most often for solution when solving an equation that is set equal to zero.
4. Equivalent equations are equations that have the same solutions.

Example The solutions to the equation x2 + 5x = 6 are the same as the roots to the equivalent equation: x2 + 5x – 6 = 0. (They both have solutions of –6 and 1.)

Types of Algebraic Equations

Conditional: An equation that is true for some values, but not others.
Identity: An equation that is true for all the values of the domain.
Contradiction: An equation that is never true.

Some examples of solving equations:
In general to solve linear equations you want to put the terms with the variable on one side of the equation and the other terms on the opposite side.
Example:
3x + 7 = 2(x - 1) - 5x
3x + 7 = 2x - 2 - 5x
3x - 2x + 5x = -2 - 7
6x = -9
x = -9/6 = -3/2

You can check your solution graphically by letting y1 = 3x + 7 and y2 = 2(x – 1) – 5x and using the intersection feature of your calculator to see where the two graphs cross.

Solving equations by factoring: The idea behind solving by factoring is when you multiply two or more factors and get zero as a result, you know that at least one of the factors must equal zero. Likewise if at least one factor is zero then a product that includes that zero factor will also be zero. Hence when you use factoring to solve an equation you must make sure the equation is set equal to zero first. Many students forget this and try to factor when the equation does not equal zero.

Example :
1 - x + x3 = x2
1 - x + x3 - x2 = 0
(1 - x) + x2(x - 1) = 0
(1 - x) - x2(1 - x) = 0
(1 - x)(1 - x2) = 0
(1 - x)(1 - x)(1 + x) = 0
(1 - x)2(1 + x) = 0     Set the factors equal to zero.
1 - x = 0 or 1 + x = 0
x = 1 or x = -1