Download Math Apps 6.2 Guided Notes

Name: ______________________________
Math Apps 6.2 Guided Notes
Solving Linear Equations
EQUATIONS
An _____________________ is a statement that two algebraic expressions are equal.
A ___________________ of an equation is a value of the variable that makes the equation a true
statement when substituted into the equation. ____________________ an equation means finding
every solution of the equation. We call the set of all solutions the ________________________, or
simply the _____________________ of an equation.
For example, x = 2 is one solution of the equation x2 – 4 = 0, because (2)2 – 4 = 0 is a true statement. But x
= 2 is not the solution, because x = – 2 is a solution as well. The solution set is actually {– 2, 2}.
EQUATIONS VS EXPRESSIONS
Note the difference between the two; equations contain an equal sign and expressions do not.
Example Problem
Determine if the given value is a solution of the equation.
(a) 4(x – 1) = 8; x = 2
(b) x + 7 = 2x – 1; x = 8
(c) 2y2 = 200; y = – 10
LINEAR EQUATIONS
A _____________________________ does not break the following rules:
1)
2)
3)
4)
Example problem
Determine which of the equations below are linear equations.
SOLVING LINEAR EQUATIONS
Solve each equation using the Addition and Subtraction Property, and check your answer.
(a) x – 5 = 9
(b) y + 30 = 110
Solve each equation using the Multiplication and Division Property, and check your answer.
SOLVING MULT-STEP LINEAR EQUATIONS
Procedures for Solving Linear Equations
Step 1
Step 2
Step 3
Example Problems
Solve the equation 5x + 9 = 29.
Solve the equation 6x – 10 = 4x + 8.
Solve the equation 3(2x + 5) – 10 = 3x – 10.
SOLVING LINEAR EQUAIONS WITH FRACTIONS
There’s a simple procedure that will turn any equation with fractions into one with no fractions at all.
You just need to find the Least Common Denominator of all fractions that appear in the equation, and
multiply every single term on each side of the equation by the LCD.
If there are any fractions left after doing so, you made a mistake!
Example Problem
2 x  3 5x x

 4
3
2 2
SOLVING FORMULAS FOR ONE SPECIFIED VARIABLE
The formula F = 95 C + 32 gives the Fahrenheit equivalent for a temperature in Celsius. Transform this
into a formula for calculating the Celsius temperature C.
CONTRADICTIONS AND IDENTITES
A __________________________ is an equation with no solution.
An _________________________ is an equation that is true for any value of the variable for which both
sides are defined.
When you solve an equation that is an identity, the final equation will be a statement that is always true.
In a contradiction the final equation will be a statement that is false.
Example Problem
Indicate whether the equation is an identity or a contradiction, and give the solution set.
(a) 3(x – 6) + 2x = 5x – 18
(b) 6x – 4 + 2x = 8x – 10