SOLUTION: If the sum of two numbers is 1, and their difference is 4, what are the two numbers?

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Question 86915: If the sum of two numbers is 1, and their difference is 4, what are the two numbers?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=1st #, y=2nd #

After translation, we get the system of equations:
x%2By=1 "the sum of two numbers is 1"
x-y=4 "their difference is 4"

Now lets solve this system by using substitution
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

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

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

y=%281-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=1-1%2Ax Now we've fully isolated y

Since y equals 1-1%2Ax we can substitute the expression 1-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-1%2Ahighlight%28%281-1%2Ax%29%29=4 Replace y with 1-1%2Ax. Since this eliminates y, we can now solve for x.

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

1%2Ax-1%2B1%2Ax=4 Multiply



1%2Ax-1%2B1%2Ax=4 Reduce any fractions

1%2Ax%2B1%2Ax=4%2B1Add 1 to both sides


1%2Ax%2B1%2Ax=5 Combine the terms on the right side



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


cross%28%281%2F2%29%282%2F1%29%29x=%285%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2%2F1 and isolate x

So when we multiply 5%2F1 and 1%2F2 (and simplify) we get



x=5%2F2 <---------------------------------One answer

Now that we know that x=5%2F2, lets substitute that in for x to solve for y

1%285%2F2%29-1%2Ay=4 Plug in x=5%2F2 into the 2nd equation

5%2F2-1%2Ay=4 Multiply

-1%2Ay=4-5%2F2Subtract 5%2F2 from both sides

-1%2Ay=8%2F2-5%2F2 Make 4 into a fraction with a denominator of 2



-1%2Ay=3%2F2 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%283%2F2%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=3%2F-2 Multiply the terms on the right side


y=-3%2F2 Reduce


So this is the other answer


y=-3%2F2<---------------------------------Other answer


So our solution is

x=5%2F2 and y=-3%2F2

which can also look like

(5%2F2,-3%2F2)

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

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

we get


graph of 1%2Ax%2B1%2Ay=1 (red) and 1%2Ax-1%2Ay=4 (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%2F2,-3%2F2). This verifies our answer.


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Check:

Plug in (5%2F2,-3%2F2) into the system of equations


Let x=5%2F2 and y=-3%2F2. Now plug those values into the equation 1%2Ax%2B1%2Ay=1

1%2A%285%2F2%29%2B1%2A%28-3%2F2%29=1 Plug in x=5%2F2 and y=-3%2F2


5%2F2-3%2F2=1 Multiply


2%2F2=1 Add


1=1 Reduce. Since this equation is true the solution works.


So the solution (5%2F2,-3%2F2) satisfies 1%2Ax%2B1%2Ay=1



Let x=5%2F2 and y=-3%2F2. Now plug those values into the equation 1%2Ax-1%2Ay=4

1%2A%285%2F2%29-1%2A%28-3%2F2%29=4 Plug in x=5%2F2 and y=-3%2F2


5%2F2%2B3%2F2=4 Multiply


8%2F2=4 Add


4=4 Reduce. Since this equation is true the solution works.


So the solution (5%2F2,-3%2F2) satisfies 1%2Ax-1%2Ay=4


Since the solution (5%2F2,-3%2F2) satisfies the system of equations


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


this verifies our answer.





Since x=5%2F2 and y=-3%2F2 this means our first number is 5%2F2 and our second number is-3%2F2