General Methodology for Solving Equations

Transcription

1 Section. Pre-Activity Preparation General Methodology for Solving Equations Catering A catering company charges $6.9 per guest with an additional set up fee of $00. How many guests can be invited if the budget is $000? Construction The cost of painting a bedroom house includes indoor paint at $9. per gallon and outdoor paint at $7.99 per gallon. If gallons of indoor paint are required, how many gallons of outdoor paint may be purchased, if the total budget for paint is $600? Forensic Science The approximate time of death* can be determined by body temperature with the formula: B N T = -. T = time since death B = body temperature at time of death N = body temperature at present If a body is found hours after death, what should the temperature be if the victim was in good health at the time of death? * The formula allows us to calculate the time since death; to find the time of death, we simply need to know the current time: if T = hours and it is :00 p.m. now, we know the time of death was 7:00 a.m. ( hours before the current time). Repair A repair company charges $ per hour plus a travel fee of $0. How many hours did the repair person work if the total charge was $90? Each of the above situations can be figured out through trial and error, or in your head, but a quicker, more direct method is to use algebra. Each of the above situations can be modeled into an algebraic equation and solved for the unknown quantity. This section provides you with the algebra skills to solve each of the these situations. Learning Objectives Be able to clear an equation of fractions or decimals Find the solution to any given algebraic equation in one variable Solve for a variable in a formula in terms of the other variables Terminology Previously Used clear parentheses formula LCD solve substitute variable New Terms to Learn clear fractions

2 Chapter Solving Equations Building Mathematical Language Solving Equations: x = a One of the main objectives of algebra courses is for students to become proficient in solving equations. Solve equations by applying the properties of equations and real numbers in successive steps until the desired variable is isolated on one side of the equal sign and its coefficient is. That is, the equation is written in the form of x = a. For algebraic equations, a will be a constant. For formulas, a can be an expression in terms for the unknown side of the other variables, such as in the equation: a = c -b of a right triangle c b Clearing Fractions Equations with fractions can be manipulated so that the fractions divide or multiply out, leaving all coefficients and constants as whole numbers. Using the multiplication property of equality, multiply both sides of an equation (with fractions) by the least common denominator (LCD) of the fractions to clear them. a Properties and Principles In order to successfully work with the equations in this section, you should remember and be able to use the Addition, Multiplication, and Distributive Properties of Equality, all of which have already been introduced. Multiplication Property Review if a = b then a : c = b : c if x =, then x = 6 Remember that dividing is also accepted under this property, because division is multiplication by the reciprocal. Therefore: if a = b then a c = b c if x =, then x = x = x =

3 Section. General Methodology for Solving Equations Methodologies Clearing Fractions from an Equation The example equations in this methodology have fractions as constants. Before isolating the variable, the fractions can be cleared by multiplying both sides of the equation by the LCD. Example : Isolate the variable in the equation: + x = Example : Isolate the variable in the equation: + x = 7 Try It! Steps in the Methodology Example Example Step Find the LCD of all the fractions in the equation. Use the methodology for finding the LCD from Foundations of Mathematics. The fractions are and. The LCM of and is their product: 0. = 0 Step Multiply both sides of the equation by the LCD. Apply the multiplication property of equations to multiply both sides of the equation by the LCD. 0 + x = 0 Step Distribute the LCD to each term. Apply the Distributive Property twice, once on the left side of the equation and once on the right side of the equation. The result is that each term in the equation is multiplied by the LCD. 0 : + 0 : x = 0 : continued on next page

4 6 Chapter Solving Equations Steps in the Methodology Example Example Step Simplify so that the fractions clear to whole numbers. Notice that the fractions clear by design; you chose the LCD for that purpose. 0 : + 0 : x = 0 + 0x = 8 0 or + 0x = 0 + 0x = 8 : Step Validate. Because each term was multiplied by the LCD (0, in this case) to arrive at the final expression, dividing each term by the LCD should result in the initial expression x 8 + = x 8 + = x = Methodologies Congratulations! Putting together what you have learned from Section. through Section., you have practiced each step of the General Methodology for Solving Equations: Step Determine the variable. Step Clear parentheses. Step Combine like terms. Step Clear fractions. Step Isolate the chosen variable. Step 6 Multiply or divide to make the chosen variable coefficient equal to. Step 7 Validate. The remaining models in this section will make use of this general methodology to solve a variety of different equations.

5 Section. General Methodology for Solving Equations 7 Model Solve: x (x + ) = x Step Determine the variable. x is the variable; it appears twice on the left side and once on the right side. Step Clear parentheses. x (x + ) = x x x = x Step Combine like terms. x = x Step Clear fractions. No fractions; omit this step. Step Isolate the variable. x x = x x Step 6 Divide. x = x + = + x = Answer: x - = x = - Step 7 Validate. ( ) x x + = x? + = 9 + =?? 9 = 9 9 =??? Why can we do this???? Why can we do this? Sometimes it minimizes the number of steps required to keep numbers as fractions rather than write them as whole numbers. Here, inside the parentheses, the expression - is equal to, but notice that is the common denominator, and each term on the left will eventually need to be written as a fraction so that it can be combined and compared to the fraction on the right. It actually adds a step to rewrite - as rather than mentally combine - with + 6.

6 8 Chapter Solving Equations Model Solve: ( x - 6) = ( x -) Step Determine the variable. The variable to solve for is x and it appears on both sides of the equal sign. Step Clear parentheses. ( x - 6) = ( x -) 8 x - = x - Step Combine like terms. No terms to combine. Step Clear fractions. LCD = 8 x = (x ) : : 8 x = 8x 8 x 8 = 8x 8 Step Isolate the variable. x -8x - 8 = 8x -8x -8 -x = x = 0 Step 6 Divide. Answer: x 0 = x = Step 7 Validate.? [( -) -6]= (-) -? (- 8 ) = ( -) - 6 = - 6 [ ] Model Solve for w: A = l + w Step Determine the variable. w is the variable; it appears once on the right side. Step Clear parentheses. No parentheses; omit this step. Step Combine like terms. No like terms to combine; omit this step. Step Clear fractions. No fractions; omit this step.

7 Section. General Methodology for Solving Equations 9 Step Isolate the variable. A - l = l - l + w Step 6 Divide. A - l = w A -l w = Answer: w = A - l (Symmetric Property of Equality) Step 7 Validate. Validate by choosing values for A and l: Let A = 00, l = 0 Substitute values into the original equation: 00 = ( 0) + w 00 = 60 + w = w 0 = w w = 0 compare Substitute values in our final equation (from Step 6): 00 -( 0) w = w = = w = 0 Model Solve:.x + 0. = x 0.(x + ) Step Determine the variable. x is the variable; it appears once on the left side and twice on the right side. Step Clear parentheses..x + 0. = x 0.x 0.6 Step Combine like terms..x + 0. = 0.7x 0.6 Step Clear fractions. No fractions; omit this step. Step Isolate the variable..x 0.7x + 0. = 0.7x 0.7x x = x =. Step 6 Divide. 0. x. = x = - 0. Answer: x =. Step 7 Validate.?. ( -. ) + 0. = (-. ) -0. (-. ) =? ( 0. ) -. =? = -. [ ]

8 0 Chapter Solving Equations Technique An Option for Step : Clearing Decimals Sometimes it is helpful or preferable to clear the decimals in an equation. Remember that we can convert decimal numbers to fractional numbers because numbers represented as decimals are fractions with denominators that are powers of 0 (0, 00, 000,...). Mentally converting decimals to decimal fractions, we can determine what power of 0 the equation should be multiplied by to clear the decimals. Clear the decimals: 0. 7x = 0. 7 Convert to decimal fractions: x = 0 00 The LCD is 00, so we need to multiply each term in the original equation by 00: 00( 0.7x) = 00( 0. ) 70x = Model Solve: a = 0. a Step Determine the variable. a is the variable; it appears once on the left side and once on the right side. Step Clear parentheses. No parentheses; omit this step. Step Combine like terms. No like terms to combine; omit this step. Step Clear decimals as in the technique above. To clear decimals, each term must be multiplied by 00: a = -00a Step Isolate the variable a + 00a = 00a + 00a 60a = a = Step 6 Divide. 60a = a = 60 Answer: a = 0.6 Step 7 Validate (0.6)? = ? = = 0.

9 Section. General Methodology for Solving Equations Addressing Common Errors Issue Incorrect Process Resolution Correct Process Validation Forgetting to multiply a term by the LCD when clearing fractions x - + = x - : + = : x - + = 6 x - = 6 x = 7 Use parentheses and distributive property when multiplying both sides of the equation by the LCD. x + = x + = ( ) x + : = : ( ) x + = 6 x + = 6 x + = 6 x = x + = ( ) +? = +? = +? = = Not validating in the original equation In the above example, validate x = 7: x + = 6 (7) + = 6 + = 6 6 = 6 Any error might be repeated in the validation if you do not use the original equation. Using the original equation will find the error. Validate x = 7 x - + = ( 7) - + = + = 8! An error has been made; rework the problem. Adding a term when it should have been subtracted or subtracting when addition is called for x 6 = 6 x = x + = x = 6 x = Even though it looks right, take the time to write in the step. x 6 = 6 x = x = 0 x + = x + = x = x x = = 0 6 =? 6 6 = 6? + =? + = =

10 Chapter Solving Equations Issue Incorrect Process Resolution Correct Process Validation Dividing incorrectly when solving for the unknown 6x = x = x = x = Even though it looks right, take the time to write in the step. 6 x 6 x = = 6 x = x : = : x = 8 6? : = 6? = = x = 8? = = Solving for the wrong variable in a formula Solve for h: A = bh A = bh bh = A A b = h After solving the equation, review Step and make sure the correct variable has been isolated. The math processes you use may be correct, but the answer is not if you have solved for the wrong variable. A = bh A = bh bh = A A h = b Preparation Inventory Before proceeding, you should have an understanding of each of the following: When and how to clear fractions in an equation How to apply the methodology of solving equations to solve an equation or formula for a given variable

11 Section. Activity Methodology for Solving Equations Performance Criteria Clear an equation of fractions appropriate common denominator is used all fractions are cleared, leaving whole number coefficients and constants an equivalent equation is formed Solve equations application of the methodology accuracy of computation appropriate validation Solve formulas (literal equations) for a given variable application of the methodology correct variable is solved in terms of the other variables all variables are accounted for in the final answer Critical Thinking Questions. If the solution to an equation is in the form x = a, what are three types of numbers that a can be?. Can x = 0 be a solution for an equation? Illustrate your answer with an example.

12 Chapter Solving Equations. Why should you clear fractions before solving an equation?. When validating an equation, why is it important use the original equation?. How does solving for a variable in a formula work if there aren t any numbers to combine? 6. What are three differences between solving an equation and solving for a variable in a formula? Tips for Success Clearing fractions (or decimals) can make it easier to work through a problem, but answers may still contain fractions or decimals Report the final answer in the same form (fraction or decimal) as the original equation In a formula, it might be helpful to highlight the chosen variable

13 Section. General Methodology for Solving Equations Demonstrate Your Understanding. Solve for x: a) x + = Problem Worked Solution Validation b) x + 6 = 0 x c) (x + ) = x d) (x 7) = (x + ) e) (x + ) 6 = 7x (x + ) f) (x + 7) = (x + 9) +

14 6 Chapter Solving Equations. Clear all fractions or decimals before solving for x: Problem Worked Solution Validation a) ( x + 6) = x - b) x x - = c) ( x + 6) + = ( x + ) d).x + 0. = x + 0.(x + ) e) 0. 0x - 0. ( 0. x + 0. ) = 0. 0x -0.

15 Section. General Methodology for Solving Equations 7. Solve for the given variable: a) Solve for w: A = l +w Problem Worked Solution Validation b) Solve for t: I = Prt c) Solve for t: D = rt d) Solve for b: x a b = + e) Solve for h: V = lwh

16 8 Chapter Solving Equations Identify and Correct the Errors In the second column, identify the error(s) in the worked solution or validate its answer. If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer. Worked Solution Identify Errors or Validate Correct Process Validation ) x x = + 6 x x = x = x + x + = x + + x = x + 7 x x = x + 7 = x x = 7 ) x = x 0 x = x x = ( ) = ( ) = 0 = ) Solve for r: F mv = r r : F = mv r = mv F ) x - = x - x : - = x - : x - = x -6 x =