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Transcript
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.