Solve the system of linear equations: 2x+y=1                                                              3x+4y=14 Please help with this. Show steps and if possible further explain the steps. I would like to learn how to properly solve this problem.

Solution:

Consider the given system of linear equations :

$$\begin{gathered} 2x + y = 1 ....\left(1\right) \\ 3x + 4y = 14 ....\left(2\right) \end{gathered}$$

For solving the given linear system we will use the elimination method, according to which we try to eliminate one variable by balancing the coefficient with multiply some integers, as

Now, eliminate 'y' from equation (1) and (2) with 

Multiplying equation (1) by 4 we have

$$\begin{gathered} 4\left(2x + y\right) = 4\times 1 \\ 8x + 4y = 4 ....\left(3\right) \end{gathered}$$

Therefore, the coefficient of 'y' in (3) and (2) is equal.

Now, subtracting equation (2) from equation (3) we have