Let's solve the problem step-by-step.
We are given the system of equations:
[tex]\[ 4m + 6n = 54 \][/tex]
[tex]\[ 3m + 2n = 28 \][/tex]
To find the determinant [tex]\( D \)[/tex] of the system, we set up the determinant of the coefficient matrix:
[tex]\[ D = \left| \begin{array}{cc} 4 & 6 \\ 3 & 2 \end{array} \right| \][/tex]
We calculate the determinant of a 2x2 matrix [tex]\( \left| \begin{array}{cc} a & b \\ c & d \end{array} \right| \)[/tex] using the formula:
[tex]\[ D = ad - bc \][/tex]
Here, [tex]\( a = 4 \)[/tex], [tex]\( b = 6 \)[/tex], [tex]\( c = 3 \)[/tex], and [tex]\( d = 2 \)[/tex].
Substituting these values into the formula, we get:
[tex]\[ D = (4 \times 2) - (6 \times 3) \][/tex]
Simplifying the expression step-by-step:
[tex]\[ D = 8 - 18 \][/tex]
[tex]\[ D = -10 \][/tex]
Therefore, the determinant [tex]\( D \)[/tex] is:
[tex]\[ D = -10 \][/tex]
