Question 173519
let the 2 numbers be a and b
then:
a + b = 64 (sum of the 2 numbers is 64)
also:
a/b = 7 (quotient of one of the numbers divided by the other number is 7)
we picked a to be the numerator and b to be the divisor.
we could have picked b to be the numerator and a to be the divisor.
the choice was ours since it was not specified which of the numbers had to be the numerator.
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you have 2 equations to work with:
a + b = 64
a/b = 7
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solve one of the variables (a and b are variables) in terms of the other.
you can use either equation but it looks like it would be simpler to use the second equation.
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a/b = 7 is the equation we are going to work with and we will solve for a in terms of b.
starting again:
a/b = 7
multiply both sides of the equation by b to get:
a = 7*b
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we go back to the first equation:
a + b = 64
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we substitute 7*b for a to get:
7*b + b = 64
we combine likes together (7*b and 1*b are likes) to get:
8*b = 64
we divided both sides of the equation by 8 to get
b = 64/8 = 8
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we have solved for b.
now we take one of the equations and solve for a.
take the second equation which is:
a/b = 7
since b = 8, this equation becomes:
a/8 = 7
multiply both sides of equation by 8 to get:
a = 7*8 = 56
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we have values for a and b:
a = 56
b = 8
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plug these values into the other equation to see if that equation holds true.
that other equation is:
a + b = 64
substituting 56 for a, and 8 for b, this equation becomes:
56 + 8 = 64
64 = 64
this equation is true.
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since both equations are true with a value of a = 56 and b = 8, these values are good and the problem is solved.
one of the numbers is 56
the other number is 8.
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note that i didn't have to prove a/b = 7 is true because i used it to solve for the second variable and, in doing so, forced it to be true.
i said:
a/8 = 7
then
a = 8*7 = 56
by solving for a, i forced that equation to be true.
to prove that, substitute anyway:
a/b = 7
substitute 56 for a, and 8 for b to get:
56/8 = 7
7=7
equation is true.
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