Chapter 1 - Fundamentals
1.5 - Equations
Definitions
Equation
An equation is a statement that two mathematical statementsare equal.
Solutions
The values of the unknown that make the equation true arecalled the solutions or roots of the equation.
1.5 - Equations
Definitions
Solving the equation
The process of finding the solutions is called solving theequation.
Equivalent Equations
Two equations with exactly the same solutions are calledequivalent equations.
1.5 - Equations
Definitions
Linear Equations
A linear equation in one variable is an equation equivalent toone of the form
where a and b are real numbers and x is the variable.
1.5 - Equations
Definitions
Literal Equations
Equations with several variables (letters) are called literalequations.
Even though there are more letters in these equations, themethods used to solve these equations are the same as themethods you use to solve all equations.
1.5 - Equations
Example 1 – pg. 55 #32
Solve the equation for the indicated variable.
1.5 - Equations
Definitions
Quadratic Equations
A quadratic equation is an equation of the form
where a, b, and c are real numbers with a 0.
1.5 - Equations
Solving Quadratic Equations
What are three methods of solving quadraticequations?
1.5 - Equations
Quadratic Formula
The roots of the quadratic equation
where a 0, are
1.5 - Equations
Example 2 – pg. 55
Solve the equations.
47.
54.
70.
1.5 - Equations
Fractional Equations
To solve fractional expressions we must
1.Factor the numerator and denominator completely.
2.State the restrictions or domain.
3.Eliminate the denominator by multiplying eachterm by the LCD.
4.Expand and combine like terms.
5.Move all terms to one side of = sign.
6.Solve the quadratic.
7.Check your solutions.
1.5 - Equations
Example 3 – pg. 55 #86
Find all real solutions of the equation.
1.5 - Equations
Radical Equations
When we solve radical equations, we must be carefulto check your final answers. We may end up with anextraneous solution, a solution that does not satisfythe original equation.
1.5 - Equations
Example 4 – pg. 55 #91
Find all real solutions of the equation.
1.5 - Equations
Algebraic Expression
An equation of the form
where W is an algebraic expression, is an equation ofquadratic type. We solve equations of quadratictype by substituting for the algebraic expression, aswe see in the next two examples.
1.5 - Equations
Example 5 – pg. 55 #97
Find all real solutions of the equation.
1.5 - Equations
Example 6 – pg. 55 #100
Find all real solutions of the equation.
1.5 - Equations
