CHAPTER R: Basic Concepts of Algebra R.1 The Real-Number System R.2 Integer Exponents, Scientific Notation, and Order of Operations R.3 Addition, Subtraction, and Multiplication of Polynomials R.4 Factoring R.5 The Basics of Equation Solving R.6 Rational Expressions R.7 Radical Notation and Rational Exponents Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

R.5 The Basics of Equation Solving Solve linear equations. Solve quadratic equations. Solve a formula for a given letter. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Equations An equation is a statement that two expressions are equal. To solve an equation in one variable is to find all the values of the variable that make the equation true. Each of these numbers is a solution of the equation. The set of all solutions of an equation is its solution set. Equations that have the same solution sets are called equivalent equations. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Equations A linear equation in one variable is an equation that is equivalent to one of the form ax + b = 0, where a and b are real numbers and a  0. A quadratic equation is an equation that is equivalent to one of the form ax2 + bx + c = 0, where a, b and c are real numbers and a  0. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Equation- Solving Principles For any real numbers a, b, and c, The Addition Principle: If a = b is true, then a + c = b + c is true. The Multiplication Principle: If a = b is true, then ac = bc is true. The Principle of Zero Products: If ab = 0 is true, then a = 0 or b = 0, and if a = 0 or b = 0, then ab = 0. The Principle of Square Roots: If x2 = k, then Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Solve. 8x = 2(18 – 2x) 8x = 36 – 4x 12x = 36 x = 3 The solution checks. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Solve. x2 – 8x = – 16 x2 – 8x + 16 = 0 (x – 4 )(x – 4) = 0 x – 4 = 0 or x – 4 = 0 x = 4 or x = 4 There is only one solution, 4. The solution checks. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Solve. The solution checks. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Formulas A formula is an equation that can be used to model a situation. We have used the motion formula, d = r • t. We have used the simple-interest formula, I = Prt. We can use the equation-solving principles to solve a formula for a given variable. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Solve P = 2l + 2w for l. Solution: The formula can be used to determine a rectangle’s length if we know its perimeter and width. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley