The sum of two numbers exceeds their product by one.Their difference is equal to their product less five. Find the two numbers.
x+y = xy+1
x-y = xy-5
so, eliminating xy,
x+y-1 = x-y+5
2y = 6
y = 3
so, that means that
x+3 = 3x+1
x = 1
check:
3+1 = 3*1+1 yes
1-3 = 3*1 - 5 yes
Where did the 2y=6come from?
x+y-1=x-y+5
x-x+y+y=5+1
2y=6
Let's call the two numbers 'x' and 'y'.
From the problem statement, we can write two equations:
Equation 1: The sum of two numbers exceeds their product by one.
x + y = xy + 1
Equation 2: Their difference is equal to their product less five.
x - y = xy - 5
To solve these equations, we can use a method called substitution.
Step 1: Rearrange Equation 2 to solve for 'x':
x - y = xy - 5
x = (xy - 5) + y
Step 2: Substitute the value of 'x' from Step 1 into Equation 1:
(x + y) = xy + 1
((xy - 5) + y) + y = xy + 1
Simplifying the equation:
2xy - 4y - 6 = 0
Step 3: Rearrange the equation to solve for 'y':
2xy - 4y - 6 = 0
2xy - 4y = 6
2y(x - 2) = 6
y(x - 2) = 3
Step 4: We can see that y = 3 is a possible solution. Now substitute y=3 into Equation 2 to solve for 'x':
x - y = xy - 5
x - 3 = 3x - 5
2 = 2x
x = 1
Therefore, the two numbers are x = 1 and y = 3.