Question

In Exercises 55–56, write the system of linear equations for which Cramer's Rule yields the given determinants.

Accepted Answer

Step 1: Understand that the determinant D corresponds to the coefficient matrix of the system of linear equations. The matrix for D is given as $$ \begin{bmatrix} 2 & -4 \ 3 & 5 \end{bmatrix} . $$This means the system has two variables, say $$ x $$ and $$ y $$, and the coefficients of these variables in the two equations are from this matrix. Step 2: The determinant $$ D_x  is $$formed by replacing the first column of the coefficient matrix with the constants from the right-hand side of the equations. The matrix for $$ D_x  is$$ $$ \begin{bmatrix} 8 & -4 \ -10 & 5 \end{bmatrix} . $$This tells us the constants on the right side of the equations are 8 and -10 respectively. Step 3: Write the system of linear equations using the coefficients from matrix D and the constants from the right side. The first equation uses the first row of D: $$ 2x - 4y = 8 . $$The second equation uses the second row of D: $$ 3x + 5y = -10 . $$Step 4: Verify that the system is consistent with the determinants given. The determinant D should not be zero for Cramer's Rule to apply, and the determinants D and $$ D_x $$ correspond to the coefficient matrix and the matrix with the first column replaced by constants, respectively. Step 5: Summarize the system of equations as: $$ \begin{cases} 2x - 4y = 8 \ 3x + 5y = -10 \end{cases} . $$This is the system for which Cramer's Rule yields the given determinants.

In Exercises 55–56, write the system of linear equations for which Cramer's Rule yields the given determinants.

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Concetti chiave

Cramer's Rule

Determinants of 2x2 Matrices

Forming Systems of Linear Equations from Determinants