Section 2.4 Applications of Linear Equations: Introduction to Problem Solving 151 10 times the first integer ° ° ¢ 22 3 times the sum of the integers ¢ 10 times the first is 3 times integer 22 more than Step 4: Write a mathematical equation. Step 5: Solve the equation. Clear parentheses. Combine like terms. 10x 331x2 1x 12 1x 224 22 10x 31x x 1 x 22 22 10x 313x 32 22 10x 9x 9 22 10x 9x 31 10x 9x 9x 9x 31 x 31 The first integer is Step 6: Interpret the results and x 31. The second integer is write the x 1 31 1 32. x 2 31 2 33. The third integer is answer in words. The three integers are 31, 32, and 33. 4. Applications of Linear Equations Using a Linear Equation in an Application Example 6 A carpenter cuts a 6-ft board in two pieces. One piece must be three times as long as the other. Find the length of each piece. Solution: In this problem, one piece must be three Step 1: Read the problem times as long as the other. Thus, if x completely. represents the length of one piece, then 3x can represent the length of the other. x represents the length of the smaller piece. Step 2: Label the unknowns. 3x represents the length of the longer piece. Draw a figure. Answer 6. One piece is 80 in. and the other is 16 in. Step 3: Write a verbal model. Isolate the x-terms on one side. ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ the sum of the integers Classroom Example: p. 155, Exercise 32 3x x Skill Practice 6. A plumber cuts a 96-in. piece of pipe into two pieces. One piece is five times longer than the other piece. How long is each piece?
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