SOLUTION: hI i have done this and would like to know if i am right . I took me 5 minutes to do it. Hope you help me. Tahnks TWo numbers differ by 3. the sum of teh larger and one fourth

Algebra ->  Problems-with-consecutive-odd-even-integers -> SOLUTION: hI i have done this and would like to know if i am right . I took me 5 minutes to do it. Hope you help me. Tahnks TWo numbers differ by 3. the sum of teh larger and one fourth      Log On


   

Question 118153: hI i have done this and would like to know if i am right . I took me 5 minutes to do it. Hope you help me. Tahnks

TWo numbers differ by 3. the sum of teh larger and one fourth the smaller is 13. What are the numbers.
Found 2 solutions by Fombitz, jim_thompson5910:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
What did you get as an answer?

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If we translate the first sentence "TWo numbers differ by 3", we get: x-y=3 and if we translate the second sentence "he sum of teh larger and one fourth the smaller is 13", we get: x%2B%281%2F4%29y=13

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


%281%29%2Ax%2B%281%2F4%29%2Ay=13 Start with the second equation


4%28%281%29%2Ax%2B%281%2F4%29%2Ay%29=%284%29%2A%2813%29 Multiply both sides by the LCD 4



4%2Ax%2B1%2Ay=52 Distribute and simplify


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Lets start with the given system of linear equations

1%2Ax-1%2Ay=3
4%2Ax%2B1%2Ay=52

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=3-1%2AxSubtract 1%2Ax from both sides

y=%283-1%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-3%2B1%2Ax Now we've fully isolated y

Since y equals -3%2B1%2Ax we can substitute the expression -3%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B1%2Ahighlight%28%28-3%2B1%2Ax%29%29=52 Replace y with -3%2B1%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax%2B1%2A%28-3%29%2B1%281%29x=52 Distribute 1 to -3%2B1%2Ax

4%2Ax-3%2B1%2Ax=52 Multiply



4%2Ax-3%2B1%2Ax=52 Reduce any fractions

4%2Ax%2B1%2Ax=52%2B3Add 3 to both sides


4%2Ax%2B1%2Ax=55 Combine the terms on the right side



5%2Ax=55 Now combine the terms on the left side.


cross%28%281%2F5%29%285%2F1%29%29x=%2855%2F1%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5%2F1 and isolate x

So when we multiply 55%2F1 and 1%2F5 (and simplify) we get



x=11 <---------------------------------One answer

Now that we know that x=11, lets substitute that in for x to solve for y

4%2811%29%2B1%2Ay=52 Plug in x=11 into the 2nd equation

44%2B1%2Ay=52 Multiply

1%2Ay=52-44Subtract 44 from both sides

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

cross%28%281%2F1%29%281%29%29%2Ay=%288%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=8%2F1 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=11 and y=8

which can also look like

(11,8)

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

1%2Ax-1%2Ay=3
4%2Ax%2B1%2Ay=52

we get


graph of 1%2Ax-1%2Ay=3 (red) and 4%2Ax%2B1%2Ay=52 (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 (11,8). This verifies our answer.


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Check:

Plug in (11,8) into the system of equations


Let x=11 and y=8. Now plug those values into the equation 1%2Ax-1%2Ay=3

1%2A%2811%29-1%2A%288%29=3 Plug in x=11 and y=8


11-8=3 Multiply


3=3 Add


3=3 Reduce. Since this equation is true the solution works.


So the solution (11,8) satisfies 1%2Ax-1%2Ay=3



Let x=11 and y=8. Now plug those values into the equation 4%2Ax%2B1%2Ay=52

4%2A%2811%29%2B1%2A%288%29=52 Plug in x=11 and y=8


44%2B8=52 Multiply


52=52 Add


52=52 Reduce. Since this equation is true the solution works.


So the solution (11,8) satisfies 4%2Ax%2B1%2Ay=52


Since the solution (11,8) satisfies the system of equations


1%2Ax-1%2Ay=3
4%2Ax%2B1%2Ay=52


this verifies our answer.






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Answer:

So the numbers are 11 and 8.