Problem 56 - College Algebra

Question

Describe the determinants \(D_{x}\) and \(D_{y}\) in terms of the coefficients and constants in a system of two equations in two variables.

Step-by-Step Solution

Answer

The determinants \(D_{x}\) and \(D_{y}\) in a system of two equations are given by \(D_{x} = ed - bf\) and \(D_{y} = af - ec\), where \(ax + by = e\) and \(cx + dy = f\) are the two equations.

Step 1: Understanding the notations

Assume we have a system of two linear equations in the form:\[ ax + by = e \]\[ cx + dy = f \]The determinant of the system \(D\) is computed from the coefficients of x and y as:\[ D = \left|\begin{array}{ll} a & b \ c & d \end{array} \right| = ad - bc \]

Step 2: Determinant \(D_{x}\)

The determinant \(D_{x}\) is computed by replacing the coefficients of x in the determinant of the system with the constants on the right side of the equations:\[ D_{x} = \left|\begin{array}{ll} e & b \ f & d \end{array} \right| = ed - bf \]

Step 3: Determinant \(D_{y}\)

Similarly, the determinant \(D_{y}\) is computed by replacing the coefficients of y in the determinant of the system with the constants on the right side of the equations:\[ D_{y} = \left|\begin{array}{ll} a & e \ c & f \end{array} \right| = af - ec \]