After completing this chapter, students will be able to: Explain the basic procedures used to solve equations for the unknown; list the five rules and the mechanical steps used to solve for the unknown in seven situations; know how to check the answers; list the steps for solving word problems; complete blueprint aids to solve word problems; check the solutions.
Solving for the Unknown: A How-To Approach for Solving Equations McGrawHill/Irwin Chapter Five Copyright © 2014 by The McGrawHill Companies, Inc. All rights reserved Learning unit objectives LU 5-1: Solving Equations for the Unknown Explain the basic procedures used to solve equations for the unknown List the five rules and the mechanical steps used to solve for the unknown in seven situations; know how to check the answers LU 5-2: Solving Word Problems for the Unknown List the steps for solving word problems Complete blueprint aids to solve word problems; check the solutions 52 Terminology Expression – A meaningful combination of numbers and letters called terms Equation – A mathematical statement with an equal sign showing that a mathematical expression on the left equals the mathematical expression on the right Formula – An equation that expresses in symbols a general fact, rule, or principle Variables and constants are terms of mathematical expressions 53 Solving Equations for the Unknown Equality in equations A+8 58 Left side of equation Right side of equation Dick’s age in years will equal 58 54 Variables and Constants Rules If no number is in front of a letter, it is a 1: B = 1B; C = 1C If no sign is in front of a letter or number, it is a +: C = +C; = +4 55 Solving for the Unknown Rule Whatever you to one side of an equation, you must to the other side 56 Opposite Process Rule If an equation indicates a process such as addition, subtraction, multiplication, or division, solve for the unknown or variable by using the opposite process 57 Opposite Process Rule A + = 58 - - A = 50 is subtracted from both sides of equation to isolate variable A on the left Check 50 + = 58 58 Equation Equality Rule You can add the same quantity or number to both sides of the equation and subtract the same quantity or number from both sides of the equation without affecting the equality of the equation You can also divide or multiply both sides of the equation by the same quantity or number (except zero) without affecting the equality of the equation 59 Equation Equality Rule 7G = 35 7G = 35 7 G = By dividing both sides by 7, G equals Check 7(5) = 35 510 Multiple Processes Rule When solving for an unknown that involves more than one process, the addition and subtraction before the multiplication and division 511 Multiple Process Rule H +2= H +2= -2 H H H () 5 -2 = Move constant to right side by subtracting from both sides = 4(3) = 12 To isolate H, which is divided by 4, we the opposite process and multiply times both sides of the equation Check 12 + = 3+2=5 512 Parentheses Rule When equations contain parentheses (which indicates grouping together), you solve for the unknown by first multiplying each item inside the parentheses by the number or letter just outside the parentheses Then you continue to solve for the unknown with the opposite process used in the equation Do the addition and subtraction first; then the multiplication and division 513 Parentheses Rule 5(P - 4) = 20 5P – 20 = 20 + 20 + 20 5P = 40 5P P = 40 = Parentheses tell us that everything inside parentheses is multiplied by Multiply by P and by Add 20 to both sides to isolate 5P on left To remove in front of P, divide both sides by to result in P equals Check 5(8 - 4) = 20 5(4) = 20 20 = 20 514 Like Unknown Rule To solve equations with like unknowns, you first combine the unknowns and then solve with the opposite process used in the equation 515 Like Unknown Rule 4A + A = 20 5A = 20 5A = 20 =4 A To solve this equation: 4A + 1A = 5A Thus, 5A = 20 To solve for A, divide both sides by 5, leaving A equals Check 4(4) +4 = 20 16 + = 20 516 Solving Word Problems for Unknowns 1) Read the entire problem 3) Let a variable represent the unknown Y = Computers Read again if necessary! 2) Ask: “What is the problem looking for?” 4) Visualize the relationship between the unknowns and variables Then set up an equation to solve for the unknown(s) 4Y + Y = 600 5) Check your results to ensure accuracy 517 Solving Word Problems for the Unknown Blueprint aid Unknown(s) Variable(s) Relationship 518 Solving Word Problems for the Unknown ICM Company sold times as many computers as Ring Company The difference in their sales is 27 How many computers of each company were sold? Unknown(s) Cars Sold: ICM Ring Mechanical Steps 4C C = 27 3C = 27 3 C= Variable(s) 4C C Relationship 4C C 27 Ring = computers ICM = 4(9) = 36 computers Check 36 - = 27 519 ... on the left equals the mathematical expression on the right Formula – An equation that expresses in symbols a general fact, rule, or principle Variables and constants are terms of mathematical... check the solutions 52 Terminology Expression – A meaningful combination of numbers and letters called terms Equation – A mathematical statement with an equal sign showing that a mathematical expression...Learning unit objectives LU 5-1 : Solving Equations for the Unknown Explain the basic procedures used to solve equations for the unknown List the five rules and the mechanical steps used