SOLUTION: if 1 is subtracted from the numerator of a fraction, the value of a resulting fraction is 1/2. However, if 7 is added to the denominator of the original fraction, the value of the

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: if 1 is subtracted from the numerator of a fraction, the value of a resulting fraction is 1/2. However, if 7 is added to the denominator of the original fraction, the value of the      Log On


   

Question 705256: if 1 is subtracted from the numerator of a fraction, the value of a resulting fraction is 1/2. However, if 7 is added to the denominator of the original fraction, the value of the resulting fraction is 1/3. Find the original fraction
Found 2 solutions by mouk, ptaylor:
Answer by mouk(232) About Me  (Show Source):
You can put this solution on YOUR website!
If fraction is +x%2Fy+

if 1 is subtracted from the numerator of a fraction, the value of a resulting fraction is 1/2.
+%28x-1%29%2Fy+=+1%2F2+
+2%28x-1%29+=+y+
+2x-2+=+y+
+2x-y+=+2+


However, if 7 is added to the denominator of the original fraction, the value of the resulting fraction is 1/3. Find the original fraction
+x%2F%28y%2B7%29+=+1%2F3+
+3x+=+y%2B7+
+3x-y+=+7+


Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

2%2Ax-1%2Ay=2
3%2Ax-1%2Ay=7

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

-1%2Ay=2-2%2AxSubtract 2%2Ax from both sides

y=%282-2%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-2%2B2%2Ax Now we've fully isolated y

Since y equals -2%2B2%2Ax we can substitute the expression -2%2B2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-1%2Ahighlight%28%28-2%2B2%2Ax%29%29=7 Replace y with -2%2B2%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-1%2A%28-2%29-1%282%29x=7 Distribute -1 to -2%2B2%2Ax

3%2Ax%2B2-2%2Ax=7 Multiply



3%2Ax%2B2-2%2Ax=7 Reduce any fractions

3%2Ax-2%2Ax=7-2 Subtract 2 from both sides


3%2Ax-2%2Ax=5 Combine the terms on the right side



1%2Ax=5 Now combine the terms on the left side.


cross%28%281%2F1%29%281%2F1%29%29x=%285%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1%2F1 and isolate x

So when we multiply 5%2F1 and 1%2F1 (and simplify) we get



x=5 <---------------------------------One answer

Now that we know that x=5, lets substitute that in for x to solve for y

3%285%29-1%2Ay=7 Plug in x=5 into the 2nd equation

15-1%2Ay=7 Multiply

-1%2Ay=7-15Subtract 15 from both sides

-1%2Ay=-8 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%28-8%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=-8%2F-1 Multiply the terms on the right side


y=8 Reduce


So this is the other answer


y=8<---------------------------------Other answer


So our solution is

x=5 and y=8

which can also look like

(5,8)

Notice if we graph the equations (if you need help with graphing, check out this solver)

2%2Ax-1%2Ay=2
3%2Ax-1%2Ay=7

we get


graph of 2%2Ax-1%2Ay=2 (red) and 3%2Ax-1%2Ay=7 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (5,8). This verifies our answer.


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Check:

Plug in (5,8) into the system of equations


Let x=5 and y=8. Now plug those values into the equation 2%2Ax-1%2Ay=2

2%2A%285%29-1%2A%288%29=2 Plug in x=5 and y=8


10-8=2 Multiply


2=2 Add


2=2 Reduce. Since this equation is true the solution works.


So the solution (5,8) satisfies 2%2Ax-1%2Ay=2



Let x=5 and y=8. Now plug those values into the equation 3%2Ax-1%2Ay=7

3%2A%285%29-1%2A%288%29=7 Plug in x=5 and y=8


15-8=7 Multiply


7=7 Add


7=7 Reduce. Since this equation is true the solution works.


So the solution (5,8) satisfies 3%2Ax-1%2Ay=7


Since the solution (5,8) satisfies the system of equations


2%2Ax-1%2Ay=2
3%2Ax-1%2Ay=7


this verifies our answer.

Answer by ptaylor(2198) About Me  (Show Source):
You can put this solution on YOUR website!
Let the fraction be x/y.
So, (x-1)/y = 1/2 multiply each term by 2y
2(x-1)=y or 2x-2=y and 2x-y=2----eq1
x/(y+7)=1/3 multiply each term by 3(y+7)
3x=(y+7)or 3x-y=7----eq2
Subtract eq1 from eq2 and we get:
x=5------------numerator
Substitute x=5 into eq1:
2*5-y=2
10-y=2
-y=-8
y=8 ----denominator
CK
(5-1)/8=4/8=1/2 ---ok
5/(8+7)=5/15=1/3---ok
Hope this helps---ptaylor