We are given a system of two linear equations in two variables, $x$ and $y$: $$4x - y = 5$$ $$3x + 2y = 1$$ We need to determine if the pairs $(1, 2)$ and $(1, -1)$ are solutions to the system, and then solve the system using the substitution method.

1. Problem Description

We are given a system of two linear equations in two variables, $x$ and $y$:

$$4x - y = 5$$

$$3x + 2y = 1$$

We need to determine if the pairs $(1, 2)$ and $(1, -1)$ are solutions to the system, and then solve the system using the substitution method.

2. Solution Steps

1) Check if $(1, 2)$ is a solution:

Substitute $x = 1$ and $y = 2$ into the equations:

$$4(1) - 2 = 4 - 2 = 2 \ne 5$$

$$3(1) + 2(2) = 3 + 4 = 7 \ne 1$$

Since the first equation is not satisfied, $(1, 2)$ is not a solution.

2) Check if $(1, -1)$ is a solution:

Substitute $x = 1$ and $y = -1$ into the equations:

$$4(1) - (-1) = 4 + 1 = 5$$

$$3(1) + 2(-1) = 3 - 2 = 1$$

Since both equations are satisfied, $(1, -1)$ is a solution.

3) Solve the system using substitution:

From the first equation, we can express $y$ in terms of $x$:

$$4x - y = 5$$

$$y = 4x - 5$$

Substitute this expression for $y$ into the second equation:

$$3x + 2(4x - 5) = 1$$

$$3x + 8x - 10 = 1$$

$$11x = 11$$

$$x = 1$$

Now, substitute $x = 1$ back into the expression for $y$:

$$y = 4(1) - 5 = 4 - 5 = -1$$

Thus, the solution is $(1, -1)$.

3. Final Answer

1) The couple $(1, 2)$ is not a solution of the system (S).

2) The couple $(1, -1)$ is a solution of the system (S).

3) The solution of the system (S) is $(1, -1)$.