Linear Diophantine Equation #4

Question

Linear Diophantine Equations with Constraints are a type of mathematical problem where you are given a set of linear equations with integer coefficients, and you are asked to find unique integer solutions that satisfy the linear equations. These puzzles are a challenging combination of mathematical reasoning and logic. Your task is to deduce the distinct values of these variables based on the provided sum equations. These puzzles challenge your ability to analyze patterns, solve equations, and manage constraints simultaneously. Try to solve them as quickly as possible, because during an interview you will be timed as well!

For all questions, you have the constraint that A – E need to be unique integers in the range of 1 – 5.

 	C + D = B + E
 	B + C = A + E
 	A - D = E - 2

Accepted Answer

Sometimes, under the pressure of time, you need to make a trade-off between solving it analytically and solving it with trial and error. In this case, a combination of both will lead to the quickest solution. First, sum up all three equations, in order to eliminate 3 out of 5 variables.

C+ D = B + E
B + C = A + E
A - D = E - 2
——————— +
2C + B + A = 3E + B + A - 2

Which we can simplify as:

 	2C = 3E - 2

Here, your quickest approach would be to try different values. Let's try out all numbers ranging from 1-5 for C.

 	If C= 1, 3E should equal 4, which is not possible with integers.
 	If C= 2, 3E should equal 6, which isn't possible, because E = 2 would result in a non-unique solution.
 	If C= 3, 3E should equal 8, which isn't possible with integers.
 	If C= 4, 3E should equal 10, which isn't possible with integers.
 	If C= 5, 3E should equal 12, which works! E = 4.

So now we have:

 	C = 5
 	E = 4

Now we can substitute these values in the first two equations:

 	5 + D = B + 4 (or D = B - 1)
 	B + 5 = A + 4 (or A = B + 1)

From this, we know that D < B < A, so:

 	D = 1
 	B = 2
 	A = 3

So the full solution is:

 	A = 3
 	B = 2
 	C = 5
 	D = 1
 	E = 4

Title	Category	Subcategory	Difficulty	Status
Linear Diophantine Equation #1	Math Questions	Solving Unknowns	Medium	Example
Linear Diophantine Equation #2	Math Questions	Solving Unknowns	Medium	Example
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