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Question 78145: One number is 3 more than 2 times the other, and their sum is 27. what are the numbers
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! The hardest part of this problem is translating the words into equations.
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The problem uses the words "one number" and "the other". This tells you that there are two
numbers to be found. Call "one number" x and "the other" y.
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The first sentence tells you that "one number" (the one we called x) is 3 more than twice
"the other" and we called that y. Twice y is 2y and 3 more than that is 2y + 3.
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So from the first sentence we know that x = 2y + 3.
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The first sentence also tells us that when we add "one number" to "the other" the sum is
27. In equation form this is x + y = 27.
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Now we have two equations as follows:
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x = 2y + 3 and
x + y = 27
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Since we have two unknowns (x and y) we must have two equations to solve for the two unknowns,
and we have our two equations.
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One of the ways to solve these two is by substitution. Notice that the first equation
tells us that x is equal to the right side and the right side has only one unknown, y.
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So we can take the right side of the first equation and plug it into the second equation
in place of x. When we do that we get:
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(2y + 3) + y = 27
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The parentheses are just there to show you that we replaced the x that was there with 2y + 3.
We can remove the parentheses to make the equation:
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2y + 3 + y = 27
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Add the terms containing y and the equation becomes:
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3y + 3 = 27
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We need to eliminate the + 3 on the left side. We can do that by subtracting 3 from the
left side, but if we do that we must also subtract 3 from the right side to keep the equation
in balance. This subtraction results in:
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3y + 3 - 3 = 27 - 3
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and this simplifies to:
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3y = 24
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Now you can solve for y by dividing both sides by 3 to get:
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3y/3 = 24/3
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which simplifies to:
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y = 8
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We now know one of the unknowns. Recall that one of our equations said that the sum of the
two unknowns must be 27. In equation form we wrote the equation:
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x + y = 27
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But we know that y is 8. So in this equation we can replace y by 8 and get:
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x + 8 = 27
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We can solve for this by subtracting 8 from both sides of the equation to get:
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x = 27 - 8 = 19
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In summary, we have found that the two unknowns are 19 and 8.
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Check ... does their sum equal 27? Yes it does. And is 19 equal to twice 8 = 3? Well, twice
8 is 16 and add 3 gives you 19 ... so this condition was met by the two numbers also.
We have found an answer using the process of substitution and have checked our answer.
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Hope this discussion helps you to see a way to work problems by substitution.
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