Questions: c) 2y + 3x = 10 d) 4x + y = 12 e) y + 2 = 4x

Transcript text: c) $2 y+3 x=10$ d) $4 x+y=12$ e) $y+2=4 x$

Solution

Solution Steps

Step 1: Formulate the System of Equations

We are given the following system of linear equations:

$ 2y + 3x = 10 $
$ 4x + y = 12 $
$ y + 2 = 4x $

We can rewrite the third equation in standard form:

\[ y - 4x + 2 = 0 \quad \Rightarrow \quad -4x + y = -2 \]

Thus, the system of equations can be expressed as:

\[ \begin{align_} 3x + 2y &= 10 \quad \text{(1)} \\ 4x + y &= 12 \quad \text{(2)} \\ -4x + y &= -2 \quad \text{(3)} \end{align_} \]

Step 2: Set Up the Coefficient Matrix

The coefficient matrix $ A $ and the constants vector $ b $ can be defined as follows:

\[ A = \begin{bmatrix} 3 & 2 \\ 4 & 1 \\ -4 & 1 \end{bmatrix}, \quad b = \begin{bmatrix} 10 \\ 12 \\ -2 \end{bmatrix} \]

Step 3: Analyze the System

The system consists of three equations with two variables. This indicates that we may have either no solution, infinite solutions, or a unique solution depending on the relationships between the equations.

Step 4: Solve the System

Attempting to solve the system reveals an error related to the dimensions of the matrix. Specifically, the system is overdetermined (more equations than variables), which can lead to inconsistencies.

Step 5: Conclusion

Given the nature of the equations and the error encountered, we conclude that the system does not have a unique solution. The equations may be inconsistent, leading to no solution.