Question 112136
The first sentence says that "one number exceeds another ..." This statement tells you that
there are two unknown numbers. Call one of them x and the other y.  Since one number 
exceeds the other by 5, you know that if you take 5 away from one of the numbers, the result
equals the other number. In equation form this can be written as:
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{{{x - 5 = y}}} <=== call this your first equation
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In the next sentence the term "of their sum" tells you that you need to add the two numbers.
Their sum is written as {{{x + y}}}. Then you need one-fifth of that sum, so you need to
multiply the sum by one-fifth. In algebraic form this is:
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{{{(1/5)(x + y)}}} <=== refer to this as the "algebraic form"
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Finally you are told that this is algebraic form is 5 less than the smaller number.
This means that if you add 5 to this algebraic form the result will equal the smaller
number. From your first equation you can see that y is the smaller of the two numbers
because you have to take 5 away from x to get y.  So add 5 to the algebraic
form and the result will equal y. In equation form this is written as:
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{{{(1/5)(x + y) + 5 = y}}} <=== call this your second equation
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You now have two independent equations, each with two unknowns. Since there is a match between
the number of equations and the number of unknowns (and the equations are independent
of each other) you can solve for each unknown.  Your first equation gives you a value for y
in terms of the second unknown ... which is x. You can see that y is equal to x minus 5.
Therefore, you can go to your second equation and in it you can replace y with x - 5. (This
method is referred to as solving the two equations by "substitution.") When you do this
substitution, the second equation becomes:
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{{{(1/5)(x + (x - 5)) + 5 = x - 5}}}
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On the left side the quantity (x + (x - 5)) can be simplified. Since the set of parentheses
that contains x - 5 is preceded by a plus sign these parentheses can be removed without
changing the signs of the x and the -5 inside. This leaves you with (x + x - 5) and when
you add the x + x you end up with (2x - 5). Don't forget that this quantity has to be
multiplied by one-fifth, so you now have reduced the left side and the equation is:
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{{{(1/5)(2x - 5)) + 5 = x - 5}}}
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You can now get rid of the {{{1/5}}} by multiplying both sides of this equation (all terms)
by 5. When you do this multiplication you get:
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{{{2x - 5 + 25 = 5x - 25}}}
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This multiplication makes it a little easier because you don't have to mess with  fractions.
On the left side combine the two numbers -5 and + 25 to get +20 and the equation then is:
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{{{2x + 20 = 5x - 25}}}
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Let's get rid of the 2x on the left side by subtracting 2x from both sides. When you do
you get:
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{{{ 20 = 3x - 25}}}
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Then get rid of the -25 on the right side by adding 25 to both sides to get:
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{{{45 = 3x}}}
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You can solve for x by dividing both sides by 3 and the result is:
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{{{15 = x}}}
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You now know that x equals 15 and your very first equation told you that if you take 5
away from x the result is y. Therefore, y equals 15 - 5 = 10.
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So the answer to this problem is x = 15 and y = 10.
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You can check the answer by returning to the two original equations, substituting 
15 for x and 10 for y, and then simplifying to ensure that the left side of the equation 
equals the right side.
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Return to the first equation and when you substitute 15 for x and 10 for y you have:
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15 - 5 = 10
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That reduces to:
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10 = 10
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so that works.
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Then go to the second equation and again substitute 15 for x and 10 for y to get:
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{{{(1/5)(15 + 10)+5 = 10}}}
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Inside the parentheses on the left side the 15 + 10 becomes 25 ... so the equation
reduces to:
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{{{(1/5)(25)+5 = 10}}}
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and one-fifth of 25 is 5 so the equation reduces to:
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{{{5 + 5 = 10}}}
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which simplifies to
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{{{10 = 10}}}
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Both your first and second equations work out if x = 15 and y = 10. Therefore, your answers
are correct.
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Hope this helps you to understand the problem and see how it can be worked out to get
the answer.
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