6.1 Order of Operations and Solving Linear Equations 301 We did not show the checking of the solution to Example 8. To save space, we will not show all checks. However, you should check all your answers when solving equations. = = x x 3 3 33 3 11 Divide both sides of the equation by 3 (division property) (Step 5). A check will show that 11 is the solution to − = x3 5 28. 7 Now try Exercise 53 Example 9 Solving a Linear Equation Solve the equation + = + − t t 6 11 2( 6) 5. Solution Our goal is to isolate the variable t. To do so, follow the general procedure for solving equations. t t t t t t t t t t t t t t t 6 11 2( 6) 5 6 11 2 12 5 6 11 2 7 6 2 11 2 2 7 4 11 7 4 11 11 7 11 4 4 4 4 4 4 1 + = + − + = + − + = + − + = − + + = + − = − = − = − = − Distributive property (Step 2) Combine like terms (Step 3). Subtraction property (Step 4) Subtraction property (Step 4) Division property (Step 5) 7 Now try Exercise 57 Did You Know? A New Concept Well into the sixteenth century, mathematicians found it difficult to accept the idea that the solution to a problem (such as Example 9) could be a negative number because negative numbers could not be accepted as physically real. In the early days of algebra, someone working a problem did not isolate a variable by subtracting like terms. Instead, a problem would be put into a form that allowed only positive coefficients and answers. Albert Girard (1595– 1637) contributed to the evolution of a correct understanding of negative quantities. Example 10 Solving an Equation Containing Fractions Solve the equation + = x2 3 1 3 3 4 . Solution When an equation contains fractions, we generally begin by multiplying both sides of the equation by the lowest common denominator, LCD (see Chapter 5). In this example, the LCD is 12 because 12 is the smallest number that is divisible by both 3 and 4. ⎛ + ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ + ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛ ⎝⎜ ⎞ ⎠⎟ x x x 12 2 3 1 3 12 3 4 12 2 3 12 1 3 12 3 4 12 2 3 12 1 3 12 3 4 4 1 4 1 3 1 Multiply both sides of the equation by the LCD (Step 1). Distributive property (Step 2) Divide out common factors. + = + − = − = = = x x x x x 8 4 9 8 4 4 9 4 8 5 8 8 5 8 5 8 Subtraction property (Step 4) Division property (Step 5) A check will show that 5 8 is the solution to the equation. 7 Now try Exercise 61
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