Question

How do you solve 4x - 3y = 8 and 2x + 5y = - 9 ?

Answer 1

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

Complete step-by-step solution:
We are given that we are required to solve 4x - 3y = 8 and 2x + 5y = - 9.
We will use substitution to solve the same.
Let us term the given equation 4x - 3y = 8 as the equation number 1 and the given equation 2x + 5y = - 9 as equation number 2.
Taking the 2x from addition in the left hand side to subtraction in right hand side in the second equation, we will then obtain the following equation:-
$ \Rightarrow $5y = - 9 – 2x
Dividing both the sides of this equation by 5, we will then obtain the following equation:-
$ \Rightarrow y = - \dfrac{1}{5}(9 + 2x)$ …………..(3)
We can now put this in equation number 1.
We will then obtain the following equation:-
$ \Rightarrow 4x - 3\left\{ { - \dfrac{1}{5}\left( {9 + 2x} \right)} \right\} = 8$
Simplifying the terms, we will then obtain the following equation:-
$ \Rightarrow 4x + \dfrac{3}{5}\left( {9 + 2x} \right) = 8$
Opening up the bracket, we will then obtain the following equation:-
$ \Rightarrow 4x + \dfrac{{27}}{5} + \dfrac{6}{5}x = 8$
Now, we will club the constant terms and the terms with x, we will then obtain the following equation:-
$ \Rightarrow \dfrac{{26}}{5}x = 8 - \dfrac{{27}}{5}$
Simplifying this further, we will then obtain the following equation:-
$ \Rightarrow \dfrac{{26}}{5}x = \dfrac{{13}}{5}$
Thus, we get: $x = \dfrac{1}{2}$
Putting this in equation number 3, we will then obtain the following equation:-
$ \Rightarrow y = - \dfrac{1}{5}\left\{ {9 + 2\left( {\dfrac{1}{2}} \right)} \right\}$
Simplifying the calculations, we will then obtain the following equation:-
$ \Rightarrow y = - 2$

Hence, the answer is $x = \dfrac{1}{2}$ and y = - 2.

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 4x - 3y = 8 …………(1) and 2x + 5y = - 9 ………(2)
Multiplying the equation number 2 by 2, then we will obtain the following equation:-
$ \Rightarrow $4x + 10y = - 18 ……….(3)
Subtracting the equation number 1 from equation number 3, we will then obtain the following equation:-
$ \Rightarrow ${4x + 10 y} – {4x – 3y} = - 18 – 8
Simplifying the equation, we will then obtain the following equation:-
$ \Rightarrow $13y = - 26
Thus, we have y = - 2
Therefore, by putting this in equation number 1, we get $x = \dfrac{1}{2}$.
Hence, the answer is $x = \dfrac{1}{2}$ and y = - 2.