Chapter 8 - XYZ Custom Plus

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518<strong>Chapter</strong> 8 Linear Equations and InequalitiesThe solution to our equation is 2. We check our work (to be sure we have notmade either a mistake in applying the properties or an arithmetic mistake) bysubstituting 2 into our original equation and simplifying each side of the resultseparately.Check: When x = 2{the equation 2(x + 3) = 10Step 4: becomes 2(2 + 3) ≟ 102(5) ≟ 1010 = 10 A true statementOur solution checks.The general method of solving linear equations is actually very simple. It isbased on the properties we developed in <strong>Chapter</strong> 1 and on two very simple newproperties. We can add any number to both sides of the equation and multiplyboth sides by any nonzero number. The equation may change in form, but thesolution set will not. If we look back to Example 1, each equation looks a little differentfrom each preceding equation. What is interesting and useful is that eachequation says the same thing about x. They all say x is 2. The last equation, ofcourse, is the easiest to read, and that is why our goal is to end up with x by itself.2. Solve for x : 2(x − 3) + 3 = 9.Example 2Solve for x : 3(x − 5) + 4 = 13.Solution Our first step will be to apply the distributive property to the left sideof the equation:Step 1a: 3x − 15 + 4 = 13 Distributive propertyStep 1c: 3x − 11 = 13 Simplify the left side3x − 11 + 11 = 13 + 11 Add 11 to both sidesStep 2:{ 3x = 24Step 3:{ 1 _3 (3x) = 1 _3 (24) Multiply both sides by 1 _3 x = 8 The solution is 8Check: When x = 8{the equation 3(x − 5) + 4 = 13becomes 3(8 − 5) + 4 ≟ 13Step 4:3(3) + 4 ≟ 139 + 4 ≟ 1313 = 13 A true statement3. Solve.7(x − 3) + 5 = 4(3x − 2) − 8Example 3Solve 5(x − 3) + 2 = 5(2x − 8) − 3.Solution In this case we apply the distributive property on each side of theequation:Step 1a: 5x − 15 + 2 = 10x − 40 − 3 Distributive propertyStep 1c: 5x − 13 = 10x − 43 Simplify each sideStep 2:5x + (−5x) − 13 = 10x + (−5x) − 43 Add −5x to both sides{−13 = 5x − 43−13 + 43 = 5x − 43 + 43 Add 43 to both sides30 = 5xAnswers1. −2 2. 6
8.4 Solving Linear Equations519Step 3:Check:Step 4:{5(3){ 1 _5 (30) = 1 _5 (5x) Multiply both sides by 1 _5 6 = x The solution is 6Replacing x with 6 in the original equation, we have5(6 − 3) + 2 ≟ 5(2 ⋅ 6 − 8) − 35(3) + 2 ≟ 5(12 − 8) − 3+ 2 ≟ 5(4) − 315 + 2 ≟ 20 − 317 = 17 A true statementNoteIt makes no differenceon which side of theequal sign x ends up.Most people prefer to have x onthe left side because we read fromleft to right, and it seems to soundbetter to say x is 6 rather than 6 isx. Both expressions, however, haveexactly the same meaning.Example 4Solve the equation 0.08x + 0.09(x + 2,000) = 690.Solution We can solve the equation in its original form by working with thedecimals, or we can eliminate the decimals first by using the multiplication propertyof equality and solving the resulting equation. Both methods follow.4. Solve the equation.0.06x + 0.04(x + 7,000) = 680Method 1Working with the decimals0.08x + 0.09(x + 2,000) = 690 Original equationStep 1a: 0.08x + 0.09x + 0.09(2,000) = 690 Distributive propertyStep 1c: 0.17x + 180 = 690 Simplify the left side0.17x + 180 + (−180) = 690 + (−180) Add −180 to each sideStep 2:{0.17x = 510Step 3:{ 0.17x _= _510 Divide each side by 0.170.17 0.17x = 3,000Note that we divided each side of the equation by 0.17 to obtain the solution. Thisis still an application of the multiplication property of equality because dividingby 0.17 is equivalent to multiplying by _10.17 .Method 2 Eliminating the decimals in the beginning0.08x + 0.09(x + 2,000) = 690 Original equationStep 1a: 0.08x + 0.09x + 180 = 690 Distributive propertyStep 1b:100(0.08x + 0.09x + 180) = 100(690) Multiply both sides by 100{ 8x + 9x + 18,000 = 69,000Step 1c: 17x + 18,000 = 69,000 Simplify the left sideStep 2: 17x = 51,000 Add −18,000 to each sideStep 3:Check:{ 17x _= 51,000_ Divide each side by 1717 17x = 3,000Substituting 3,000 for x in the original equation, we have0.08(3,000) + 0.09(3,000 + 2,000) ≟ 690{0.08(3,000) + 0.09(5,000) ≟ 690Step 4:240 + 450 ≟ 690690 = 690 A true statementAnswers3. 0 4. 4,000
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