Question 462112: Find three consecutive odd integers such that four times the middle integer is equal to two less than the sum of the first and third integers.
Found 2 solutions by solver91311, math-vortex:
Answer by solver91311(24713)   (Show Source):
Answer by math-vortex(648)  (Show Source):
You can put this solution on YOUR website! Find three consecutive odd integers such that four times the middle integer is equal to two less than the sum of the first and third integers.
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This is a great problem to practice translating English sentences into mathematical sentences and algebraic expressions.
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Let
a = the first integer
b = the middle integer
c = the middle integer
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We can write "four times the middle integer" as 4b.
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"The sum of the first and third integers" may be written as a + c.
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"Two less than the sum of the first two integers" may be written as (a + c) - 2.
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Putting it all together, "four times the middle integer is equal to two less than the sum of the first and third integers," may be written as the equation:
4b = (a + c) - 2
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Now we have a single equation, but three variables. We will use the fact that the integers are consecutive, odd integers to write expressions for b and c in terms of a.
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The integer b is in the middle so it is two more than a, so
b = a + 2
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The integer c is the largest so it is two more than a.
c = a + 4
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Now we take our equation and substitute a + 1 for b and a + 2 for c:
4b = (a + c) - 2
4(a + 2) = (a + (a + 4)) -2
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Now combine like terms and solve for a.
4a + 8 = a + a + 4 - 2
4a + 8 = 2a + 2
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Subtract 2a from both sides, and simplify.
4a + 8 - 2a = 2a + 2 - 2a
2a + 8 = 2
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Subtract 8 from both sides, and simplify.
2a = -6
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Divide both sides by 2.
a = -3
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Since a, b, and c are consecutive intergers with a the smallest, the numbers are -3, -1, and 1.
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Let's check our work using the original English sentences. "Four times the middle integer is equal to two less than the sum of the first and third integers."
4 times -1 is -4.
The sum of -3 and 1 is -2.
-4 is 2 less than -2. (TRUE!)
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Hope this helps!
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Ms.Figgy
math.in.the.vortex
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