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CHAPTER 7 Linear Equations
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unadd 5 That is, subtract 5 from both sides of the equation (equivalently, add 5 to both sides of the equation) In short, what is added must be subtracted; what is subtracted must be added; what is multiplied must be divided; and what is divided must be multiplied There are other operation pairs (an operation and its inverse); some will be discussed later In much of this book, when the coe cient of x (the number multiplying x) is an integer, both sides of the equation will be divided by that integer And when the coe cient is a fraction, both sides of the equation will be multiplied by the reciprocal of that fraction
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5x 2 5x 2 5 5 2 x 5 2 x 6 3 3 2 3 x 6 2 3 2 x 9 3x 18 3 18 x 3 3 x 6 x 2 x 2 1 1 (Remember that x can be written 1x; 1 is its own reciprocal) Divide both sides by 5 or multiply both sides by
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1 x 7 4 1 x 4 7 4 4 x 28
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(Remember, reciprocals have the same sign)
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CHAPTER 7 Linear Equations
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1: 4x 36 2: 2x 26 3: 4: 3 x 24 4 1 x 5 3
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1: 4x 36 4 36 x 4 4 x 9 2: 2x 26 2 26 x 2 2 x 13 3: 3 x 24 4 4 3 4 x 24 3 4 3 x 32 1 x 5 3 1 x 3 5 3 3 x 15
Some equations can be solved in a number of ways However, the general method in this book will be the same:
CHAPTER 7 Linear Equations
1 Simplify both sides of the equation 2 Collect all terms with variables in them on one side of the equation and all nonvariable terms on the other (this is done by adding/ subtracting terms) 3 Factor out the variable 4 Divide both sides of the equation by the variable s coe cient (this is what has been factored out in step 3) Of course, you might need only one or two of these steps In the previous examples and practice problems, only step 4 was used In the following examples, the number of the step used will be in parentheses Although it will not normally be done here, it is a good idea to verify your solution in the original equation
Examples
2 x 3 7 5x 8 2x 6 7 5x 8 1 2x 1 5x 8 1 2x 2x 1 3x 8 2 8 8 9 3x 2 9 3 x 4 3 3 3 x 3 2x 1 2 10 x 1 6x 3 2 10 x 1 1 6x 5 9 x 1 x x 2 7x 5 9 5 5 2 7x 14 7 14 x 4 7 7 x 2
CHAPTER 7 Linear Equations
Practice
1: 2x 16 10 2: 3 x 1 5 2
3: 6 8 2x 25 5 2 3x 4: 4 8 3x 2x 8 5: 7 2x 3 4 x 5 8 x 1 3 6: 1 6x 8 3 x 2 4 2x 1 2
7: 5x 7 6 x 2 4 2x 3 8: 3 2x 5 2 4x 1 5 x 3 2 9: 4 3x 2 x 6 5x 8
Solutions
1: 2x 16 10 16 16 2x 6 2 6 x 2 2 x 3 2: 3 x 1 5 2 1 1 3 x 6 2 2 3 2 x 6 3 2 3 x 4
CHAPTER 7 Linear Equations
3: 6 8 2x 25 5 2 3x 48 12x 25 10 15x 73 12x 10 15x 73 73 12x 63 15x 15x 15x 3x 63 3 63 x 3 3 x 21 4: 4 8 3x 2x 8 32 12x 2x 8 2x 2x 32 10x 8 32 32 10x 40 10 40 x 10 10 x 4 5: 7 2x 3 4 x 5 8 x 1 3 14x 21 4x 20 8x 8 3 10x 41 8x 5 8x 8x 2x 41 5 41 41 2x 36 2 36 x 2 2 x 18