Answer

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**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 $y = 8$.

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