Question

Solve the following systems of equations:

$$\begin{gathered} 152x-378y=-74 \\ -378x+152y=-604 \end{gathered}$$

Answer 1

Step-1 :Adding of linear equations:

$$\begin{gathered} 152x–378y=-74\dots .\left(1\right) \\ -378x+152y=-604\dots .\left(2\right) \end{gathered}$$
Adding equation $\left(1\right)$ and equation $\left(2\right)$, we obtain
$$\begin{gathered} 152x–378y–378x+152y=-74–604 \\ \Rightarrow -226x–226y=-678 \\ \Rightarrow -226(x+y)=-678 \\ \Rightarrow x+y=\frac{678}{226} \\ \Rightarrow x+y=3\dots .\left(3\right) \end{gathered}$$

Step-2: Subtracting of linear equations:
Subtracting equation $\left(2\right)$ from equation $\left(1\right)$, we obtain
$$\begin{gathered} 152x–378y–(-378x+152y)=-74–(-604) \\ \Rightarrow 152x–378y+378x–152y=-74+604 \\ \Rightarrow 530x–530y=530 \\ \Rightarrow 530(x–y)=530 \\ \Rightarrow x–y=1\dots .\left(4\right) \end{gathered}$$

Step-3 : Solve new obtained linear equations:
Now, adding equation $\left(3\right)$ and equation $\left(4\right)$, we obtain,
$$\begin{gathered} x+y+x–y=3+1 \\ \Rightarrow 2x=4 \\ \Rightarrow x=2 \end{gathered}$$
Substituting $x=2$ in equation $\left(3\right)$,
$$\begin{gathered} 2+y=3 \\ \Rightarrow y=1 \end{gathered}$$
Hence, $\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}$ and $\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}$ are the solution of given equation.