Question 207593: Solve each system using substitution
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1.the sum of two numbers is 27. one number is 3 more than the other. find the numbers
2.find the two numbers whose sum is 58 and whose difference is 16.
3.the difference between two numbers is 18. twice the smaller number plus 3 times the larger is 74. what are the numbers?
Answer by mickclns(59)   (Show Source):
You can put this solution on YOUR website! Let's try #3 first.
State what your variables are and what they stand for: x and y are two numbers, x is the larger.
Translate the sentences into math:
x - y = 18 (left side: difference of 2 nums, =: is, 18: 18)
2y + 3x = 74 (twice the smaller: 2y, plus: +, three times the larger: 3x, is: =, 74: 74)
If neither of your math sentences has a variable and nothing else on one side, change it to an equivalent sentence which does (in this case, you probably don't want to do that with the second sentence, because whether you solve for x or y, you will have a fraction to deal with -- sometimes both sentences are like that.
x - y = 18 adding y to both sides will leave x by itself on the left
x = 18 + y So, for this problem x always stands for the same number as 18 added to the number y stands for. In particular it , x, stands for 18 + y in the equation 2y + 3x = 74. so you can SUBSTITUTE one for the other so you get an equation that has y's and numbers, so you can solve for y:
2y + 3(18+y) = 74 multiply out
2y + 54 + 3y = 74 combine like terms on the left, then subtract 54 from both sides
5y = 20 divide both sides by 5
y = 4 if all we wanted was the smaller number, we'd be done. But the question asks what are the numberS? (always go back to the word problem to make sure you have answered the question)
x = 18 + (4) = 22.
So the smaller of the two numbers is 4 and the larger is 22.
Now check IN THE WORD PROBLEM, not in the original math equations. If you translated wrongly, your answers might work just fine, but not work in the word problem.
Is the difference between the two number 18? Yes, 22 - 4 is 18.
Is twice the smaller number plus 3 times the larger, 74? Yes, 2(4) + 3(22) is 8 + 66 = 74.
Now, 1 and 2 should be easier to do.
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