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MTH 070 Elementary Algebra Chapter 2 Equations and Inequalities in One Variable with Applications 2.1 – Properties of Equality and Linear Equations Copyright © 2010 by Ron Wallace, all rights reserved. Three Algebraic Processes Changing expressions into simpler equivalent expressions. Changing equations/inequalities into simpler equivalent equations/inequalities. Solving equations/inequalities. Primary Primary (i.e. only) Tools? Tools? The algebraic properties. “Don’t make stuff up.” Review of Basic Properties Identities (0 & 1) These are used to find Inverses simpler Opposites: a+(-a) = 0 equivalent Reciprocals: b*(1/b) = b/b = 1 expressions. Commutative x + 0 = x & 1x = x Order of calculations (+ or *) Associative Grouping of calculations (+ or *) Distributive Solving Equations Solution or Root : A value for the variable (or set of values for the variables) that makes the equation a true statement. How many solutions? Finding Solutions: Guess! Estimate Graphically Algebraically 2x 5 9 x y 5 x 1 5 2 This chapter … Linear Equations w/ 1 Variable With grouping symbols removed … Linear terms Constant terms Example: 7x Example: 12 One variable Possibly more than once. 3( x 2) 9 4 x 2x 5 9 x y 5 x 1 5 2 Solving Algebraically Basic Strategy … Linear Equations w/ 1 Variable Determine simpler “equivalent equations.” Goal: x = ? Tools … Arithmetic Basic Properties Addition Property of Equality Multiplication Property of Equality Expressions VS. Equations Expressions only involve constants, variables, operators, & grouping symbols. Simplify & Evaluate An Equation is a statement that two expressions are equal. Solve 3( x 2) 9 4 x 3( x 2) 9 4x Symmetric Property of Equality a b Symmetric Property of Equality b a Symmetric Property of Equality a b b Switching sides of an equation produces an equivalent equation. a Addition Property of Equality a b Addition Property of Equality ac b Addition Property of Equality Any expression may be added to both sides of an equation without changing its solutions. ac b c Solving: Add the opposite of equation. Why? 5 x+5=7 + (-5) = 0 5 to both sides of the and x+0=x Similar forms … 5+x=7 7=x+5 x–5=7 5–x=7 Solving: x + [something] = [something else] A different but equivalent viewpoint … A term may be moved to the other side of an equation by changing its sign. x+5=7 x=7–5 x=2 Solving: x – [something] = [something else] A different but equivalent viewpoint … A term may be moved to the other side of an equation by changing its sign. x–5=7 x=7+5 x = 12 Multiplication Property of Equality a b Multiplication Property of Equality ac b Multiplication Property of Equality Both sides of an equation may be multiplied by any non-zero expression without changing its solutions. ac bc Why? Solving: Multiply both sides of the equation by the reciprocal of 5. or … divide both sides by 5. Why? 5 5x = 15 (1/5) = 1 and 1x = x Similar forms … x 15 5 2 3 x5 2x 5 3 5x 15 Checking a Solution In the original equation, replace all occurrences of the variable by the solution. Note: Put the solution in parentheses. Independently determine the value of each side of the equation. i.e. Do the arithmetic. Use a calculator (esp. TI-84). Compare the results … are they the same? ALWAYS check your solutions! Using Both Properties 2x 7 15 2x 12 5x 2x 7 x 5 2 x 7 7 5x Example 1 of 2 Applications A rhombus is a 4-sided figure with all sides having the same length. In Europe there is a rhombus-shaped sign that indicates a priority road. Find the length of each side of the priority road sign if its perimeter is 243.84 centimeters. Example 2 of 2 Applications Dean Silver has been offered two commission-based jobs selling motorcycles for two local dealers. One dealer is offering 2% commission on total monthly sales plus $1,800 per month, while the other dealer is offering 4% commission on total monthly sales plus $1,200 per month. Solve the equation 0.02s + 1800 = 0.04s + 1200, where s represents total monthly sales, to determine how much Dean must sell so that the wages of the two jobs are equal.