Isolating a Variable

Transcription

1 Section 2.3 Pre-Activity Preparation Isolating a Variable How long? How fast? How far? Base equation: d = rt (d = distance, r = rate, t = time) You are on a road trip with friends. It is your turn to drive the next 3 hours. You have your cruise control set at 70 mph. The state line is only 220 miles away. Will you still be behind the wheel when you see the Welcome! signs? d = rt d = (3)(70) = 210 miles (sorry, not this time) The next day, you and your friends all agree that stopping for dinner at a famous steak house on the way to the hotel sounds great. It is 4:00 p.m. and the reservations you have made are for 7:30 p.m. The restaurant is 200 miles away. How fast will you have to drive to make it there on time? r d = r = (200) t (3.5). 57 miles per hour (good thing the speed limit is 60 miles per hour!) Dinner was great, but you only have 2 hours left to make it to your hotel before midnight. Because of congestion and road construction, the speed of traffic has dropped to 40 miles per hour. The hotel is 85 miles away. Will you make it there in time, or should you phone the hotel to let them know you will be late? t d = t = (85) r (40) = hours (you will need to pull over and make a phone call) The same base equation was used in each case, but it was manipulated so that the variable we needed to solve for was isolated on one side of the equals sign. Learning Objectives Expanding the tool set for manipulating and transforming equations Validate that transformations are correct Learn to isolate a variable Terminology Previously Used coefficient exponent like terms terms variable New Terms to Learn isolate a variable 127

2 128 Chapter 2 Solving Equations Building Mathematical Language Simplifying Expressions Expressions are in their simplest form when there are no parentheses to remove and when all like terms have been combined. Simplify expressions by applying the Distributive Property to remove parentheses and then use the Methodology for Combining Like Terms from Chapter 1. 3x + 7y (x + 3y) = 3x + 7y 1(x + 3y)??? Why can we do this? = 3x + 7y x 3y Distributive Property??? Why can we do this? = 3x x + 7y 3y Commutative Property = (3x x) + (7y 3y) Associative Property = (3 1)x + (7 3)y Distributive Property = 2x + 4y A negative sign in front of parentheses within an expression indicates subtraction. Subtraction is addition of the opposite of a number. Inserting the coefficient of one helps highlight the distribution of the negative sign to each term in the expression. Therefore, when subtracting an expression within parentheses, multiply the expression by negative one. Isolating a Variable Variables can appear on both sides of the equal sign such as in the equation: 3x + 5 = x 7 Each side of the equation is simplified, but the variable is present on both sides. Isolating the variable is the process of applying the Addition Property of Equality to get the variable on just one side of the equal sign so that the equation can be solved. Solving equations is a process of applying the principles and properties of equations to rewrite the equation in successive equivalent forms until the solution is reached. The methodology for solving equations is presented in Section 2.4. Isolating the variable is step 3 of the methodology. The methodology for isolating a variable has some of the same steps used in solving an equation.

3 Section 2.3 Isolating a Variable 129 Properties and Principles In order to successfully work with the equations in this section, you should remember and be able to use the Distributive Property, which has already been introduced. Distributive Property Review 1. Distribute multiplication over addition (remove parentheses): 5(x + y) = 5x + 5y 2(x + y) = 2x 2y 3(x y) = 3x + 3y 2. Remove common factors from two or more terms: 3x + 6y = 3(x + 2y) x y = 1(x + y) = (x + y) 2a + 6b = 2(a 3b) 3. Combine like terms: 4x + 7x = x(4 + 7) = (4 + 7)x = 11x 5y 3 9y 3 = (5 9)y 3 = 4y 3 Models Model 1 Simplify the expression: 3(x 5y) 4(2x 3y) 3(x 5y) 4(2x 3y) = 3x 15y 8x + 12y Distributive Property = 3x 8x 15y + 12y Commutative Property = (3x 8x) + ( 15y + 12y) Associative Property = (3 8)x + ( )y Distributive Property = 5x + ( 3)y Answer: = 5x 3y To validate an expression, you can select a value for the variable and substitute it into both the original expression and your answer (the simplified version). Compare those values. Note that this method will only work when validating an expression, not an equation. To validate the work above, let x = 2 and y = 3: original expression 3( x -5y) -4( 2x -3y) 3(( 2) -5( 3)) -4( 2( 2) -3( 3)) 3( 2-15) -4( 4-9) 3( -13) -4(-5) = -19 simplified expression -5x -3y -5( 2) -3( 3) = -19

4 130 Chapter 2 Solving Equations Model 2 Simplify the expression: 3x 2 5x + 4 (2x 2 + 7x 9) 3x 2 5x + 4 (2x 2 + 7x 9) = 3x 2 5x + 4 1(2x 2 + 7x 9) (Remember we can insert a coefficient of 1 and distribute the 1 in the next step.) = 3x 2 5x + 4 2x 2 7x + 9 Distributive Property = 3x 2 2x 2 5x 7x Associative Property Answer: = x 2 12x + 13 Distributive Property Let x = 3: original expression simplified expression 2 2 3x - 5x + 4 -( 2x + 7x -9) 2 2 3( 3) - 5( 3) + 4 -( 2( 3) + 7( 3) -9) 3( 9) - 5( 3) + 4 -( 2( 9) + 7( 3) -9) ( ) = -14 x 2-12x + 13 ( ) ( ) = -14 Methodologies Isolating a Variable Example 1: Isolate the variable in the equation: 3(4x 6) = 2(x + 2) Example 2: Isolate the variable in the equation: 2(3x + 7) = 3(x + 5) + 3 Try It! Steps in the Methodology Example 1 Example 2 Step 1 Observe the equation, noting where the variables and constants occur. The equation has variables on both sides of the equal sign. Each side must be simplified by applying the Distributive Property. Special Case: For equations that contain only variables, treat all variables except the one being isolated as constants. See Model 4. The variables are highlighted and the constant factors to be distributed are shown. variables 3(4x 6) = 2(x + 2) constant factors

5 Section 2.3 Isolating a Variable 131 Steps in the Methodology Example 1 Example 2 Step 2 Choose the variable to isolate. In this case, there is only one variable, x, so there is no choice. If the equation has more than one variable, after choosing the variable, treat all other variables like constants. Step 3 Remove parentheses and combine like terms. In simplifying expressions with the Order of Operations, parentheses are evaluated first; for equations, first remove parentheses by applying the Distributive Property. 3(4x 6) = 2(x + 2) 12x + 18 = 2x + 4 Step 4 Move or collect the variables to one side of the equal sign; combine like terms. Use the Addition Property of Equality to add like quantities to both sides so that the term you want to move adds to zero or cancels out on one side and adds to the other side. Recall that adding the opposite is the same as subtracting. 12x 2x + 18 = 2x 2x x + 18 = 4 You might choose, instead, to align the terms like this: 12x + 18 = 2x + 4 2x = 2x 14x + 18 = 4 Step 5 Move or collect the constants to one side of the equal sign and combine like terms. Do the same for the constants. The equation is ready to solve. 14x = x = 14 You can align the terms like this: 14x + 18 = 4 18 = 18 14x = 14 Step 6 Validate. Make sure the equation looks like: (coefficient)(chosen variable) =constant (or simplified expression). Keep track of the arithmetic, especially the arithmetic of signed numbers. Work backwards through Steps 5, 4, 3, and 2. Alternatively, rework the problem after a short break. 14x = = x + 18 = 4 2x = 2x 12x + 18 = 2x + 4 3(4x 6) = 2(x + 2) 3(4x 6) = 2(x + 2)

6 132 Chapter 2 Solving Equations Model 3 Isolate the variable in the equation: 3x (x + 5) = 7 2(x 4) Step 1 Variables are on both sides of the equation, the Distributive Property is needed to remove parentheses, and constants are on both sides. Step 2 Variable is x. Step 3 Remove parentheses and combine like terms. 3x 1(x + 5) = 7 2(x 4) 3x x 5 = 7 2x + 8 2x 5 = 15 2x Step 4 Move variables to one side and combine like terms. 2x 5 + 2x = 15 2x + 2x 4x 5 = 15 Step 5 Collect constants and combine like terms. 4x = Answer: 4x = 20 Step 6 Validate. 4x = 20 2x 5 = 2x 5 2x 5 = 2x + 15 Model 4 Isolate the variable x: y = mx + b Step 1 The equation has all variables and no constants; therefore, treat all but the chosen variable like constants. Step 2 Choose variable x. Step 3 Step 4 No parentheses, omit this step. x is on one side of the equation. Step 5 Collect constants. y b = mx + b b Answer: y b = mx or mx = y b Step 6 Validate. y b = mx + b = + b y = mx + b??? Why can we do this? Alternatively, validate by choosing values for y, m, and b: Let y = 7, m = 3, b = 4 Substitute values into the original equation: 7 = 3x = 3x 3x = 3 x = 1 compare Substitute values in our final equation (from Step 5): 3x = 7-4 3x = 3 x = 1??? Why can we do this? Remember that the Symmetric Property of Equations states that if a = b, then b = a.

7 Section 2.3 Isolating a Variable 133 Addressing Common Errors Issue Incorrect Process Resolution Correct Process Validation Multiplying instead of adding to combine like terms x + 2x = 3x 2 Understand that the Distributive Property is used for combining like terms. Use parentheses to factor out the like variable or constants and then combine the like terms within parentheses. x + 2x = = (1 + 2)x = 3x Another way to validate an expression is to substitute a value and compare the answer with the original expression: Let x = 5. (5) + (2 5)? = ? = But: 5 + (2 5)? = ? = = 15 Subtracting only the first term of an expression within parentheses 5 (2x + 3) = =5 2x + 3 = 2x + 8 Insert a 1 in place of the negative sign to remind you that both terms are subtracted. Distribute the 1 to each term within parentheses. 5 (2x + 3) = = 5 1(2x + 3) = 5 + ( 1)2x + ( 1)3 = 5 2x 3 = 2x + 2 Let x = 10 is 2(10) +2 the same as 5 [(2 10) + 3)]? = 18 and 5 23 = 18 Leaving off a term that does not combine with any other term (forgetting a term) 3x + 2y 1 + x 3y = = 4x y Scan through the expression and determine how many different terms there are, including variables and constants. Highlight, underline, circle or mark off each term as it is combined. Every term has to be accounted for, even if it adds up to zero. 3x + 2y 1 + x 3y = x variables: 3x + x y variables: 2y 3y constants: 1 = 3x + x + 2y 3y 1 = 4x y 1 Check the terms of your solution against the original problem. Make sure that all variables are present (in this case, x and y), as well as constants (terms without a variable). x, y, constant

8 134 Chapter 2 Solving Equations Issue Incorrect Process Resolution Correct Process Validation Adding a term when it should have been subtracted or the reverse (sign error) 2x 3 = 12 + x 3x 3 = 12 3x = 9 Write in the step of showing the addition process. The term to move must add out to zero (cancel out) on one side. 2x 3 = 12 + x 2x x 3 = 12 + x x x 3 = 12 x = x = ? = ? = = 17 Not distributing to all terms 2(x + 1 y) = 3 2x y = 3 Understand that the process distributes the multiplication over addition; each addition term must be multiplied by the factor being distributed. 2 (x + 1 y) = 3 2x 2 + 2y = 3 Dropping a negative sign 5 2x = x = 7 5 2x = 2 Use parentheses to keep your signs and terms together. Even though the sign in front of the 2x does indicate subtraction, once the 5 and 5 add out, that subtraction sign still stands. Changing it to addition of 2x will help you remember. 2x = ( 2x) = 7 5 2x = 7 5 2x = ( 2x) = 7 5 2x = 2 Preparation Inventory Before proceeding, you should have an understanding of each of the following: Use of the Distributive Property to combine like terms Use of the Distributive Property to remove parentheses Use of the Addition Property of Equality to isolate a variable

9 Section 2.3 Activity Isolating a Variable Performance Criteria Application of the Distributive Property each term is included or accounted for terms are combined as needed accuracy of calculations Isolate a variable in preparation for solving an equation appropriate variable is isolated steps are validated Critical Thinking Questions 1. What does it mean to isolate a variable? 2. If you are given the formula, A = lw where A = area, l = length, and w = width, provide an example of three ways in which you can make use of isolating a variable. (Hint: review the scenarios in the introduction to this section.) 3. How do you ensure that you have isolated a variable correctly? 135

10 136 Chapter 2 Solving Equations 4. What can you do to ensure that your signs are correct? 5. When validating by substituting a value for the variables, how do you know what value to choose? 6. When you distribute across parentheses, how many terms should you end up with? 7. What properties are used for isolating a variable?

11 Section 2.3 Isolating a Variable 137 Tips for Success Experts often do more than one step at a time when isolating a variable. Work until you can combine the two steps of adding variables and constants into the same step. Combine like terms whenever you can. The result is a simpler equation with fewer symbols and thus fewer chances for error. Demonstrate Your Understanding 1. Use the Distributive Property to simplify the following: a) 3(x + y) Problem Worked Solution Validation b) 2(x + y) c) 2(x y) d) 5(x 2 2y) 2. Combine like terms to simplify the following expressions: a) 7x 2 + 5x 2 Problem Worked Solution Validation b) 3x x c) 6x 2 + 7x + 3x + x 2

12 138 Chapter 2 Solving Equations Problem Worked Solution Validation d) x x + 3x x + 7 e) 3x + 5x 2 x 4x + x 3. Use the Distributive Property first, then combine like terms to simplify the following: Problem Worked Solution Validation a) 4(n + 3) 2(n 5) b) 2(3x + 6y 1) + 5(x 3y) c) 2(x 1) + 4(x + 7) d) 4(x 2 + 2) 2(x 7) e) 2(6x 1) + 5(x 2) f) (x + 2) (x 2)

13 Section 2.3 Isolating a Variable Isolate the variable for the following equations. Do not solve. a) 9 5x = 12 Problem Answer Validation b) x + 4 = 2x + 19 c) 7x 3 = x d) 2(3x + 5) = 3x 35 e) (4x 17) = 3(x + 3) f) 2(x + 1) 4 = 7x 3(x + 4) g) 2 (3x + 7) = 4(x + 3) + 3 h) 2(2x + 5) = x 15 i) 3(4x 7) = 3(x + 6)

14 140 Chapter 2 Solving Equations 5. Isolate for the given variable: Problem Worked Solution Validation a) Isolate for x: 5x + 2y = 0 b) Isolate for y: 5x + 2y = 0 c) Isolate for C: F = 1.8C + 32 d) Isolate for x: 2x + 3y = x 7y e) Isolate for y: 2x + 3y = x 7y

15 Section 2.3 Isolating a Variable 141 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 1) Combine like terms: 2x 2 5y + 7 y + 3x 2 3y 2x 2 + 3x 2 5y y 3y 5x 2 9y 2) Isolate x: 3x 2(x + 1) = 7 3x 2x + 1 = 7 x + 1 = 7 x = 6 3) Isolate t: 5(t 2) = 3(t 1) 5t + 10 = 3t 3 3t 3t 2t + 10 = 3 10 = 10 2t = 13 4) Isolate x: y = m(x + 7) y = mx 7m y + 7m = mx

16 142 Chapter 2 Solving Equations Worked Solution Identify Errors or Validate Correct Process Validation 5) Simplify and combine like terms: 2(x 1) + 5x 2x 2 + 5x 7x 2 2