Question

Which combination of integers can be used to generate the Pythagorean triple (5,12,13)

Answer 1

Answer:

$x=3$ and $y=2$

Step-by-step explanation:

The Pythagorean triples can be generated by two values x, y, and a given system of equations:

$x^{2}-y^{2}=5\\2xy=12\\x^{2}+y^{2}=13$

You can see that each coordinate of the triple is included in each equation.

Remember that Pythagorean triples refers to the values of each side of a right triangle, where is used the Pythagorean Theorem. But, at a higher level, to construct this triples we use the system of equations, with two integers x and y., like this case.

Now we solve the system, the best first step is to just sum the first and third equations, because they have like terms:

$2x^{2}=18\\x^{2}=\frac{18}{2}=9\\x=3$

Now, we just replace it in the second equation:

$2xy=12\\y=\frac{12}{2x}=\frac{6}{3}=2$

Therefore the integers that generate the Pythagorean triple $(5,12,13)$ are $x=3$ and $y=2$

Answer 2

Answer:

x=3 and y=2

Step-by-step explanation:

The pythagorean triples are generated by two integrers x and y that can be found by solving the following system of equations:

$\left \{ {{x^{2}-y^{2}=5}\atop {2xy=12}} \atop {x^{2}+y^{2}=13}}\right.$

Solve the system of equations, and we get that the solution is x=3 and y=2.

Therefore, the combination of integrers that ca be used to generate the pythagorea triple are: x=3 and y=2