Question

$27x + 31y = 85 \\ 31x + 27y = 89$
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Answer 1

$\sf \huge{Method \ one} \\ \sf{The \ substitution \ method}$

Given,

27x + 31y = 85

→ 27x = 85 - 31y

$\sf\implies x = \frac{85 - 31y}{27}$

And, 31x + 27y = 89

So, by substituting the value of x in terms of y, we get :-

$\sf 31\left(\frac{85 - 31y}{27}\right)+ 27y = 89 \\ \\ \sf \implies \frac{2635 - 961y}{27} + \frac{729y}{27} = 89 \\ \\ \sf \implies \frac{2635 - 232y}{27} = 89 \\ \\ \sf \implies 2635 - 232y = 89 \times 27 \\ \\ \sf \implies 2635 - 232y = 2403 \\ \\ \sf \implies 232y = 2635 - 2403 \\ \\ \sf \implies 232y = 232 \\ \\ \sf \implies y = 1$

Now, since y = 1, x = (85 - 31y)/27

→ x = (85 - 31)27

→ x = 54/27

→ x = 2

$\sf \huge{Method \ two} \\ \sf{The \ elemination \ method}$

Given 27x + 31y = 85

and, 31x + 27y = 89

Multiplying with 27 in first equation and 31 in second equation, we get

27(27x + 31y) = 27(85)

→ 729x + 837y = 2295

and 31(31x + 27y) = 31(89)

→ 961x + 837y = 2759

Now subtract equation 1 from second

→ 961x + 837y - (729x + 837y) = 2759 - 2295

→ 961x + 837y - 729x - 827y = 464

→ 232x = 464

→ x = 464/232

→ x = 2

Substitute the value of x in any equation and you will get the value of y as 1.

$\huge\mathfrak{Answer} \\ \sf \huge x = 2 \\ \sf\huge y = 1$

Answer 2

Answer: x =2 and y=1

Step-by-step explanation:

27x +31 y =85 -----(1)

31 x + 27y =89 ----(2)

Now Multiplying the coefficient of x to both the equations so that we get the similar coefficient of x.

(27x +31y =85) × 31

(31x + 27y =89) ×27

After solving these equations, the result we got comes out to be -

837x + 961y = 2635

837x + 729y = 2403

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__________________

            232 y = 232

                    y = $\frac{232}{232}$

y = 1

Now substituting this value of y=1 in equation 1 , we get

27x +31y =85

$\implies$27x +31×1 =85

$\implies$27x + 31 =85

$\implies$27x =85-31

$\implies$27x =54

$\implies$ x=$\frac{54}{27}$

x= 2

So the values of x and y respectively are 2 and 1 .

I have applied elimination method here because it seems to me the most convinient, but as per your will you can also use other method like substitution, which is used by another user @mankuthemonkey01 .

Hope it helps :)