Section 2.1 Solving Linear Equations Involving Two Operations

Section 2.1 Solving Linear Equations Involving Two OperationsObjectivesIn this section, you will learn to: To successfully complete this section,you need to understand:Define the term equation.Apply the Addition Property of Equality.Apply the Multiplication Property of Equality.Isolate variables.Solve equations involving one operation.Solve equations involving two operations. The reciprocal (R.3)Additive identity (1.1)Multiplicative identity (1.1)Adding and subtracting realnumbers (1.3 and 1.4)Multiplying and dividing realnumbers (1.5)The main operation (1.10) INTRODUCTIONThe equal sign ( ) is used primarily two different ways in mathematics. In Chapter 1, we usedcontinued equal signs when simplifying an expression to indicate that one step is equivalent the nextstep.This is illustrated in Example 1 from Section 1.7, simplifying an expression using the order ofoperations:(4 – 16) 22Subtract first. -12 22Apply the exponent -12 4Divide. -348We also sometimes see continued equal signs when simplifying a fraction, such as 72 . We might needto divide out some smaller common factors before it is completely simplified:4872 Simplify by afactor of 22436 46 Simplify by afactor of 62 3Simplify by afactor of 2An equal sign is also used to indicate the equivalence of two expressions that may or may not containvariables. This equivalence is called an equation. The equations that we see in this text includenumbers and variables, and we are required to find the value of the variable that makes the equationtrue. These equations are often true for only one or two numbers.2.1 Solving Equations Involving Two Operations159 Robert H. Prior, 2014Copying is prohibited.

For example, the equation 5 x 7 is true only when x 2.We can verify this by replacing x with 2 and evaluating each side:5 (2) 77 7 True!The equation x 7 2x 1 is true only when x 6.(6) 7 2(6) 1We can verify this by replacing each x with 6 and evaluating each side:13 12 113 13 True!Some equations, called identities, are true for all numbers.For example, the equation x 3 x 3 is true whenx 2:(2) 3 (2) 35 5 True x 10:x -7:(10) 3 (10) 3(-7) 3 (-7) 313 13 True -4 -4 True Because x 3 x 3 is true for all values of x, the equation is an identity.Equations that have only one or two solutions are called conditional equations. The equationsintroduced at the beginning of this section, 5 x 7 and x 7 2x 1 , are conditional equationsbecause they each have only one solution.The discussion in this section is about solving conditional equations by finding the value of the variablethat makes an equation true.PREPARING TO SOLVE LINEAR EQUATIONSThere is a class of conditional equations called linear equations. In a linear equation, the highest powerof any variable is 1. Here are some examples of linear equations:(a)3x 5 23(b)-9 15 – 2y(c)6w 11 -4w – 9To be well prepared to solve linear equations, we must be familiar with the following ideas fromChapter 1.2.1 Solving Equations Involving Two Operations160 Robert H. Prior, 2014Copying is prohibited.

SectionPropertyExamplea 0 a and 0 a a9 0 91.The additive identity is 0.1.12.The multiplicative identity is 1.1.13.For addition: the sum of anumber and its opposite is 0.1.3a (-a) 04 (-4) 04.For multiplication: the product ofa number and its reciprocal is 1.1.5a bb · a 1344 · 3b·1 band 1 · b b1 · 15 15-2-55 · 2If a number is negative, itsreciprocal is also negative.5.The sign of a term.1.8 1 1The sign in front of a term belongs In the expression3w – 4, the secondto that term.term is -4.6.If a variable is in the numeratorof a fraction, the coefficient ofthe variable is the whole fraction,not just the numerator.1.8axac c x7.The main operation.1.10The main operation of anexpression is the last operation tobe applied, according to the orderof operations.3x34 4 xThe coefficient of x3is 4 .In the expression6y 5, the mainoperation is addition.THE VOCABULARY OF EQUATIONSAn equation is a mathematical sentence in whichone expression another expressionEach expression has one or more terms, and each term is either a constant orcontains a variable. Any term that contains a variable is called a variableterm.Note:If the highest exponent of the variable is 1, such as x1 or x, then the equation isa linear equation. A linear equation typically has only one solution.At right are some examples of linear equations. Noticethat the variable terms can be on the left side, on the rightside, or on both sides:2.1 Solving Equations Involving Two Operations161Linear is pronouncedlih -nee-er.a)n 3 -8b)18 6vc)6x 5 xd)64 y 178 – 2y Robert H. Prior, 2014Copying is prohibited.

Each of these equations is a conditional equation and is true for only one value. To solve an equation isto find the one number that makes the equation true; this number is called the solution.Throughout this section, and the next two, we look at avariety of strategies for solving linear equations. If amistake is made during the solving process, then we canstill get an answer, but it won’t be the correct answer. Inother words, it won’t be the solution.After we solve an equation and arrive at an answer, wecan verify if the answer is, indeed, the solution byplacing the answer into the equation, as demonstratedin Example 1.Example 1:Note:In the process of determining if areplacement value is a solution, we use the?symbol until we can verify whether ornot the last statement is true.For each equation, a student solved the equation and arrived at the answer shown.Verify whether the answer is a solution or not.a)12 8 – 4x; x 5b)3w – 2 2w – 5; w -3Procedure:For each equation, replace the variable with the given answer and evaluate eachside. If the two sides are equal, then the answer makes the equation true, and it isthe solution. If they are not equal, then the answer is not the solution.Answer:a)12 8 – 4xReplace x with 5.?12 8 – 4(5)?12 8 – 2012 -12 Falseb)3w – 2 2w – 5No, 5 is not the solution.Replace each w with -3.?3(-3) – 2 2(-3) – 5?-9 – 2 -6 – 5-11 -11 True 2.1 Solving Equations Involving Two Operations162Yes, -3 is the solution. Robert H. Prior, 2014Copying is prohibited.

You Try It 1a)For each equation, a student solved the equation and arrived at the answer shown.Verify whether or not the answer is a solution. Use Example 1 as a guide.15 – 5m 30; m 3b)18 5p – 2; p 4c)6v 3 4v – 1; v -2BALANCING EQUATIONSThe strategy for solving a linear equation is to isolate the variable. To isolate the variable means to getthe variable alone on one side of the equation and a single number on the other, such as x -5 or 3 y.For example, to isolate x in the equation x 3 8, we must remove the constant, 3, fromthe left side. This will leave x alone (isolated) on that side.In the solving process, as we work our way toward isolating the variable, we must often write severalequations, each one a bit simpler than the previous one. This is called reducing the equation. Eachequation in this series of reduced equations is equivalent to the others, and these equivalent equationsall have the same solution.To create simpler, reduced, equations we make changes to the expressions in the equation. To maintainthe solution, though, we must keep the equations balanced by performing the same operation—addition,subtraction, multiplication, or division—on each side of the equation.This idea of keeping an equation balanced can be visualized using a scale.Here we see the equation x 3 8. The leftside, x 3, and the right side, 8, are equal to eachother, so they are in balance.To isolate the variable, we might want to removethe 3 from the left side by subtracting it away.However, if we subtract 3 from only one side,then that side becomes lighter and the scale (andthe equation) is out of balance.2.1 Solving Equations Involving Two Operations163x 38x 3–38 Robert H. Prior, 2014Copying is prohibited.

If we subtract 3 from each side of the scale at thesame time, then the scale remains in balance andwe maintain a balanced equation.x 3–38–3In the example above, subtracting 3 from each side creates a constant term of 0 on the left side, therebyisolating the variable:x 3 8Subtract 3 from each side.x 3–3 8–3Simplify each sidex 0 5Simplify the left side.x 5The solution is 5.Although the solution itself is 5, it is common to state it as x 5.In general, to isolate the variable or the variable term, we create 0 by adding the opposite of any term wewant to remove, or clear. The property that allows us do this is the Addition Property of Equality:The Addition Property of EqualityWe may add any number or term, c, to each side of an equation,and the equality remains:Ifthena b,a c b cWhen clearing the constant, it is important to keep inmind the notion that the sign in front of the termbelongs to that term.This helps us to identify the sign of the constant term,and we can then add its opposite to each side of theequation.2.1 Solving Equations Involving Two Operations164Note:As we see in Example 2b), it is commonto write the final equation with thevariable on the left and the constant onthe right.This will be most beneficial in Section2.7, Solving Inequalities, so we should getin the habit now of placing the variableon the left side in the answer. Robert H. Prior, 2014Copying is prohibited.

Example 2:Solve each equation by clearing the constant. Also, verify the solution.28x – 6 4b) -9 4 yc) w – 5 5a)Procedure:Apply the Addition Property of Equality. Recognize the constant that must becleared, and add its opposite to isolate the variable. Verify the solution by using itas a replacement value for the variable in the original equation.Answer:a)x – 6 4x – 6 6 4 6Clear the constant -6 byadding 6 to each side.Verify the solution, 10:x–6 4?10 – 6 Simplify each side; -6 6 0.44 4b)x 0 10x 0 is just x.x 10Verify x 10-9 4 y-9 (-4) 4 (-4) y Clear the constant 4 byadding -4 to each side.Verify the solution, -13:-9 4 y?-9 Simplify each side; 4 (-4) 0.4 (-13)-9 -9-13 0 y0 y is just y.-13 yWrite the equation with thevariable on the left side.y -13c)28w – 5 5 Verify y -13 2Clear the constant - 5 by2adding 5 to each side.10Verify the solution, 2 or 5 :28w – 5 52282w – 5 5 5 510 2 ? 5 –5Simplify each side;22- 5 5 0.10w 0 5w 22.1 Solving Equations Involving Two Operations85858 5 Simplify the right side.10To verify the solution, use w 5165 Robert H. Prior, 2014Copying is prohibited.

Look back at Example 2 and notice these three things:1. Each step in the solving process is directly below the preceding step.2. The equal signs are lined up, one below the other.3. We isolated the variable by creating 0, the additive identity, on the side of the variable term.You Try It 2Solve each equation by clearing the constant. Also, verify the solution. Use Example2 as a guide.a)x 7 20b)-19 b – 12c)w – 9 -9d)837 m 3ANOTHER ADDITION TECHNIQUEAnother technique for clearing a constant is to add the constant’s opposite to each side below theconstant term. For instance, in Example 2a), in the equation x – 6 4, we cleared the constant, -6, byadding it’s opposite, 6, to each side: x – 6 6 4 6.The other technique, though, allows us to write the 6 directly below the constants on each side:x – 6 4 6 6Notice the equal sign between 6’sthat we are adding to each side.x 0 10Notice that the equal signs are allaligned, one below the other.x – 6 4 6 6x 0 10The equal sign between the 6’s is required to show that we are adding 6 to each side. This isespecially important when the equations have more terms, as in Section 2.2.Example 3 demonstrates this technique, “adding the opposite” below the term we are intending to clear.2.1 Solving Equations Involving Two Operations166 Robert H. Prior, 2014Copying is prohibited.