Question

How do you solve the following system? $8x - 5y = 9$ and $5x - 7y = 2$

Answer 1

Hint: We will find the value of y from the second equation and then put it in the first equation. After that we will get the value of x and then put that in the second equation to get the value of y.

Complete step by step solution:
We are given that we are required to solve the system of equations $8x - 5y = 9$ and $5x - 7y = 2$.
We will use substitution to solve the same.
Let us term the given equation $8x - 5y = 9$ as the equation number 1 and the given equation $5x - 7y = 2$ as equation number 2.
Consider $5x - 7y = 2$:
Re – arranging the terms, we will get:-
$ \Rightarrow y = \dfrac{{5x - 2}}{7}$ ………………(3)
Putting this value in equation number 1, we will then obtain the following expression with us:-
$ \Rightarrow 8x - 5\left( {\dfrac{{5x - 2}}{7}} \right) = 9$
Simplifying the left hand side of the above expression, we will then obtain the following equation:-
$ \Rightarrow 8x - \dfrac{{25x}}{7} + \dfrac{{10}}{7} = 9$
Simplifying the left hand side of the above expression further, we will then obtain the following equation:-
$ \Rightarrow \dfrac{{56x - 25x}}{7} + \dfrac{{10}}{7} = 9$
Taking $\dfrac{{10}}{7}$ from addition in the left hand side of the above expression to subtraction in the right hand side, we will then obtain the following expression:-
$ \Rightarrow \dfrac{{31x}}{7} = 9 - \dfrac{{10}}{7}$
Simplifying this further, we will then obtain the following equation:-
$ \Rightarrow \dfrac{{31x}}{7} = \dfrac{{53}}{7}$
Thus, we get: $x = \dfrac{{53}}{{31}}$
Putting this in equation number 3, we will then obtain the following equation:-
$ \Rightarrow y = \dfrac{{5\left( {\dfrac{{53}}{{31}}} \right) - 2}}{7}$
Simplifying the calculations, we will then obtain the following equation:-
$ \Rightarrow y = \dfrac{{29}}{{31}}$

Hence, the answer is $x = \dfrac{{53}}{{31}}$ and $y = \dfrac{{29}}{{31}}$.

Note:-
The students must note that you may use alternate methods for solving the equations other than using the substitution method.
Alternate Way:
We are given that we are required to solve $8x - 5y = 9$ …………(1) and $5x - 7y = 2$ ………(2)
Multiplying the equation 1 by 7 and 2 by 5, we will then obtain the following equations respectively:
$ \Rightarrow 56x - 35y = 63$ ……………(3)
$ \Rightarrow 25x - 35y = 10$ ……………(4)
Subtracting the equation number 4 from equation number 3, we will then obtain the following equation:-
$ \Rightarrow \left\{ {56x - 35y} \right\} - \left\{ {25x - 35y} \right\} = 63 - 10$
Simplifying the equation, we will then obtain the following equation:-
$ \Rightarrow 31x = 51$
Thus, we have $x = \dfrac{{53}}{{31}}$
Therefore, by putting this in equation number 2, we get $y = \dfrac{{29}}{{31}}$.
Hence, the answer is $x = \dfrac{{53}}{{31}}$ and $y = \dfrac{{29}}{{31}}$.