Question

Solve the following systems of equations-
$
  3x - \dfrac{{y + 7}}{{11}} + 2 = 10 \\
    \\
  2y + \dfrac{{x + 11}}{7} = 10 \\
$

Answer 1

Hint- Simplifying both the equations given in the question will make the answer easier. We will find out the value of y in terms of x first and then use the method of substitution in order to find the value of x and y both.

Complete step-by-step answer:
First, we will simplify the given equations one by one-
Marking the equations given above as equation 1 and equation 2-
$ \Rightarrow 3x - \dfrac{{y + 7}}{{11}} + 2 = 10$ (equation 1)
$ \Rightarrow 2y + \dfrac{{x + 11}}{7} = 10$ (equation 2)
Considering equation 1, we will have-
$ \Rightarrow 3x - \dfrac{{y + 7}}{{11}} + 2 = 10$
$
   \Rightarrow \dfrac{{33x - y - 7}}{{11}} = 8 \\
    \\
   \Rightarrow 33x - y - 7 = 88 \\
    \\
   \Rightarrow 33x - y - 95 = 0 \\
$
Considering equation 2, we will have-
$ \Rightarrow 2y + \dfrac{{x + 11}}{7} = 10$
$
   \Rightarrow 14y + x + 11 = 70 \\
    \\
   \Rightarrow x + 14y - 59 = 0 \\
$
Let the following equation be equation 3-
$ \Rightarrow 33x - y - 95 = 0$ (equation 3)
And this equation be equation 4-
$ \Rightarrow x + 14y - 59 = 0$ (equation 4)
Now, we will find the value of y in terms of x from equation 3-
$
   \Rightarrow 33x - y - 95 = 0 \\
    \\
   \Rightarrow y = 33x - 95 \\
$
Substituting this value of y into equation 4-
$
   \Rightarrow 14\left( {33x - 95} \right) + x = 59 \\
    \\
   \Rightarrow 462x - 1330 + x = 59 \\
    \\
   \Rightarrow 463x = 59 + 1330 \\
    \\
   \Rightarrow 463x = 1389 \\
    \\
   \Rightarrow x = \dfrac{{1389}}{{463}} \\
    \\
   \Rightarrow x = 3 \\
$
Putting this value of x in $y = 33x - 95$, we get-
$
   \Rightarrow y = 33 \times 3 - 95 \\
    \\
   \Rightarrow y = 4 \\
$
Thus, the value of x and y is $x = 3,y = 4$.

Note: Method of substitution is the most basic method to be used in such types of questions. Another method is elimination. Remember to find out the value of y as before it was in terms of x. then substitute it.