Question

The value of x and y satisfying the two equations$32x + 33y = 31$, $33x + 32y = 34$ respectively will be
(a)-1, 2
(b)2, -1
(c)0,0
(d)2,3

Answer 1

Hint – There can be two methods to solve this problem the first one can be the substitution method and the other one can be the elimination method. The second method will be discussed in the later part of this solution. In the first method of substitution, simply consider any one equation and find the relationship of one variable with the other variable, then simply substitute this variable in terms of another variable in the other equation, this helps getting an equation with only one variable. Solve to get this variable and using the relationship obtained previously, solve for the other variable as well. This will help approach the solution to this problem.

Complete step-by-step answer:
Given equations having variable x and y are
$32x + 33y = 31$.............................. (1)
And $33x + 32y = 34$...................... (2)
Substitution method:
In this method we have to evaluate any variable from any equation in the term of another variable and substitute this value in another equation which is unused and simplify this equation so this is all about the substitution method.
Now we solve these equations by substitution method so we have from equation (1) by taking 33y into R.H.S so we have,
$ \Rightarrow 32x = 31 - 33y$
Now divide by 32 throughout we have,
$x = \dfrac{{31 - 33y}}{{32}}$...................... (3)
Now substitute this value of x in equation (2) we have,
$ \Rightarrow 33\left( {\dfrac{{31 - 33y}}{{32}}} \right) + 32y = 34$
Now multiply by 32 throughout we have,
$ \Rightarrow 33\left( {31 - 33y} \right) + {\left( {32} \right)^2}y = 34\left( {32} \right)$
Now simplify this we have,
$ \Rightarrow 1023 - 1089y + 1024y = 1088$
$ \Rightarrow 1023 - 65y = 1088$
$ \Rightarrow 65y = 1023 - 1088 = - 65$
Now divide by 65 throughout we have,
$ \Rightarrow y = \dfrac{{ - 65}}{{65}} = - 1$
Now substitute this value in equation (3) we have
$ \Rightarrow x = \dfrac{{31 - 33\left( { - 1} \right)}}{{32}} = \dfrac{{31 + 33}}{{32}} = \dfrac{{64}}{{32}} = 2$
So the values of x and y satisfying the two equations are (x, y) = (2, -1).
So this is the required answer.
Hence option (B) is the correct answer.

Note – In the second method that is the method of elimination we simply make the coefficients of any one variable same for both the equations and then by application of some basic arithmetic operations like addition/subtraction we eliminate this variable and thus an equation is obtained in a single variable only, solve it to get the value of the variable. The value of the other previously eliminated variable can be obtained by simply equating any one of the two equations given.