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How do you solve $7x+2y=-19$ and $-x+2y=21$ ?

seo-qna
Last updated date: 20th Jul 2024
Total views: 382.2k
Views today: 6.82k
Answer
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382.2k+ views
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 $7x+2y=-19$ and $-x+2y=21$.
We will use substitution to solve the same.
Let us term the given equation $7x+2y=-19$ as the equation number 1 and the given equation $-x+2y=21$ as equation number 2.
Taking the $7x$ from addition in the left hand side to subtraction in right hand side in the first equation, we will then obtain the following equation:-
$ \Rightarrow 2y=-19–7x$
Dividing both the sides of this equation by 2, we will then obtain the following equation:-
$ \Rightarrow y = - \dfrac{1}{2}(19 + 7x)$ …………..(3)
We can now put this in equation number 2.
We will then obtain the following equation:-
$ \Rightarrow - x + 2\left\{ { - \dfrac{1}{2}\left( {19 + 7x} \right)} \right\} = 21$
Simplifying the terms, we will then obtain the following equation:-
$ \Rightarrow - x - \left( {19 + 7x} \right) = 21$
Opening up the bracket, we will then obtain the following equation:-
$ \Rightarrow - x - 19 - 7x = 21$
Now, we will club the constant terms and the terms with x, we will then obtain the following equation:-
$ \Rightarrow - 8x = 40$
Simplifying this further, we will then obtain the following equation:-
$ \Rightarrow x = - 5$
Thus, we get: $x = - 5$
Putting this in equation number 3, we will then obtain the following equation:-
$ \Rightarrow y = - \dfrac{1}{2}\left\{ {19 - 7\left( 5 \right)} \right\}$
Simplifying the calculations, we will then obtain the following equation:-
$ \Rightarrow y = 8$

Hence, the answer is $x = - 5$ and $y = 8$.

Note: Alternate Way:
We are given that we are required to solve $7x + 2y = - 19$ …………(1) and $- x + 2y = 21$ ………(2)
Subtracting the equation number 1 from equation number 2, we will then obtain the following equation:-
$ \Rightarrow \left\{ { - {\text{ }}x + 2y} \right\}-\left\{ {7x + 2y} \right\} = 21-\left( { - 19} \right)$
Simplifying the equation, we will then obtain the following equation:-
$ \Rightarrow - 8x = 40$
Thus, we have $x = - 5$
Therefore, by putting this in equation number 1, we get $y = 8$.
Hence, the answer is $x = - 5$ and $y = 8$.