To determine whether the system of equations is independent, dependent, or inconsistent, we can use the method of comparing the coefficients and solving for the determinants.
The system given is:
1. [tex]\( 12x + 3y = 12 \)[/tex]
2. [tex]\( y = -4x + 5 \)[/tex]
First, let's rewrite both equations in standard form:
1. [tex]\( 12x + 3y = 12 \)[/tex]
2. [tex]\( 4x + y = 5 \)[/tex] (Rewriting the second equation by moving all terms to one side)
Now we have the equations in the form of:
[tex]\[
\left\{
\begin{array}{ccccc}
a_1 x & + & b_1 y & = & c_1 \\
a_2 x & + & b_2 y & = & c_2
\end{array}
\right.
\][/tex]
Where:
- [tex]\( a_1 = 12 \)[/tex], [tex]\( b_1 = 3 \)[/tex], [tex]\( c_1 = 12 \)[/tex]
- [tex]\( a_2 = 4 \)[/tex], [tex]\( b_2 = 1 \)[/tex], [tex]\( c_2 = 5 \)[/tex]
Next, we compute the determinant of the coefficient matrix:
[tex]\[
\Delta = \begin{vmatrix}
a_1 & b_1 \\
a_2 & b_2
\end{vmatrix}
= \begin{vmatrix}
12 & 3 \\
4 & 1
\end{vmatrix}
= (12 \cdot 1) - (3 \cdot 4)
= 12 - 12 = 0
\][/tex]
Since the determinant [tex]\(\Delta\)[/tex] equals 0, the system's coefficient matrix is singular, meaning the equations are either dependent or inconsistent.
To further determine between dependent or inconsistent, we check the augmented matrix determinant:
[tex]\[
\Delta' = \begin{vmatrix}
a_1 & c_1 \\
a_2 & c_2
\end{vmatrix}
= \begin{vmatrix}
12 & 12 \\
4 & 5
\end{vmatrix}
= (12 \cdot 5) - (12 \cdot 4)
= 60 - 48 = 12
\][/tex]
Since the determinant of the augmented matrix [tex]\( \Delta' \)[/tex] is not equal to 0, the system has no solution, meaning it is inconsistent.
Therefore, the system of equations given is inconsistent.
