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ALGEBRA II. NOTES
SECTION 1.7 SOLVING EQUATIONS IN ONE VARIABLE
Definitions:
1.) Open Sentence – an equation or inequality that contains a variable.
Ex. 5x  3  18 or 2 x  6  0
2.) Solution/ Root – any value of the variable that makes an open sentence true.
- satisfies the open sentence
Ex. Find the solution for the previous examples: 5x  3  18 or 2 x  6  0
3.) Solution Set – set of all solutions of the sentence that belong to a given domain of the
variable and make the sentence true.
Note: Unless otherwise stated, our domain will always be the set of real #’s ( ) .
4.) Equivalent Equations – equations that have the same solution set over a given
domain.
Ex. 5x  3  18 and x  3 are equivalent equations.
How do we solve equations in one variable?
GOAL: Get the variable by itself!
In order to do this, we need to transform the equation.
Steps to Solve for the Variable:
1.) Simplify each side of the equation as needed.
2.) If the side containing the variable involves a certain order of operations, apply the
inverse operations in the opposite order.
Note: To get the variable by itself, we use INVERSE OPERATIONS.
1.) If adding a #, we subtract the same # from both sides.
2.) If subtracting a #, we add the same # to both sides.
3.) If multiplying by a nonzero #, we divide by the same # on both sides.
4.) If dividing by a nonzero #, we multiply by the same # on both sides.
Ex.1.) 5x  3  18
Ex.2.) y  5  5  y
Ex.3.) 3(2 x  3)  6( x  1)  10
Note: There are 3 types of solutions that can occur:
1.) Exactly 1 solution.
2.) More than 1 solution.
3.) No Solution.
Sample Problems:
1.) 3(2 x  5)  7( x  9)
2.) 5h  2(4  h)  5  3(1  h)
3.) Solve for z : w 
az  b
cz  d
5.) Formula – an equation that states the relationship between 2 or more variables.