Problem 66 - College Algebra

Question

Consider the system $$ \begin{array}{l}a_{1} x+b_{1} y=c_{1} \\\a_{2} x+b_{2} y=c_{2}\end{array} $$ Use Cramer's rule to prove that if the first equation of the system is replaced by the sum of the two equations, the resulting system has the same solution as the original system.

Step-by-Step Solution

Answer

Applying Cramer's rule to the original and the modified system, the determinants calculated for both systems will turn out to be identical. Which implies that the solutions for \(x\) and \(y\) remain the same even if the first equation of the original system is replaced by the sum of the two equations.

Step 1: Formulate the original and modified system as matrices

Let's write our original system \(a_{1}x + b_{1}y = c_{1}\) and \(a_{2}x + b_{2}y = c_{2}\) in matrix form as \(AX = C\). After replacing the first equation by the sum of the two equations, the modified system becomes \((a_{1}+a_{2})x + (b_{1}+b_{2})y = (c_{1}+c_{2})\) and \(a_{2}x + b_{2}y = c_{2}\), it can be written in matrix form as \(A'X = C'\).

Step 2: Apply Cramer's rule to the original system

According to Cramer's rule, the solutions for \(x\) and \(y\) in the original system can be given as \(x = \det(A_{x})/\det(A)\) and \(y = \det(A_{y})/\det(A)\). Where, \(A_{x}\) is obtained from \(A\) by replacing the first column by \(C\) and \(A_{y}\) is obtained from \(A\) by replacing the second column by \(C\). Calculate these values.

Step 3: Apply Cramer's rule to the modified system

Similarly, the solutions for \(x\) and \(y\) for the modified system can be given as \(x = \det(A_{x}')/\det(A')\) and \(y = \det(A_{y}')/\det(A')\). Where, \(A_{x}'\) is obtained from \(A'\) by replacing the first column by \(C'\) and \(A_{y}'\) is obtained from \(A'\) by replacing the second column by \(C'\). Calculate these values.

Step 4: Compare the solutions

We prove that the above solutions from the original and modified systems are indeed the same by demonstrating that the determinants involved are equivalent. This means showing \(\det(A_{x})/\det(A) = \det(A_{x}')/\det(A')\) and \(\det(A_{y})/\det(A) = \det(A_{y}')/\det(A')\).

Key Concepts

Understanding Systems of Linear Equations

Systems of linear equations involve finding the values of variables that satisfy multiple linear equations simultaneously. In a two-variable system, like the one in the exercise, we have two equations involving the same variables, typically denoted as \(x\) and \(y\). Each equation can be represented as a line on a coordinate plane, and the solution to the system is the point where these lines intersect.

• The first equation is \(a_{1}x + b_{1}y = c_{1}\).
• The second equation is \(a_{2}x + b_{2}y = c_{2}\).

When solving such systems, we aim to find the specific values of \(x\) and \(y\) that make both equations true simultaneously. Using methods like substitution, elimination, or Cramer's Rule, we can solve these systems efficiently. In matrix form, the system can be expressed as \(AX = C\), where \(A\) is the matrix of coefficients, \(X\) is the vector of variables, and \(C\) is the vector of constants.

Exploring Matrix Determinants

Determinants are a special number that can be calculated from a square matrix. They provide useful properties about the matrix and are essential in methods like Cramer's Rule for solving linear systems. The determinant of a 2x2 matrix, for example, \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), is calculated as \(ad - bc\).

• If the determinant is not zero, it indicates that the matrix is invertible and the system has a unique solution.
• If it is zero, the matrix doesn't have an inverse, suggesting the system might have infinitely many solutions or no solution.

In Cramer's Rule, determinants are used to find the values of variables in a system of equations. Specifically, the determinant of the original coefficient matrix and altered matrices, where one column is replaced with the constant terms, are compared to determine the solution of the system.

An Algebraic Proof Using Cramer's Rule

Cramer's Rule offers a straightforward algebraic method for solving systems of linear equations with an equal number of equations and unknowns, using determinants. To apply Cramer's Rule, we first need to ensure that the determinant of the coefficient matrix is non-zero, confirming that a unique solution exists.

• We replace each column of the coefficient matrix \(A\) in turn with the constants vector \(C\), to create matrices \(A_x\) and \(A_y\).
• The solutions are then \(x = \frac{\det(A_x)}{\det(A)}\) and \(y = \frac{\det(A_y)}{\det(A)}\).

In the exercise, we compared the solutions from the original and modified systems. This involved demonstrating that the determinants for both systems correspond, hence proving algebraically that the solutions are indeed identical. The key behind this proof is showing the equivalence of the systems through the preservation of their determinant properties after modification, illustrating the beauty and efficiency of algebraic methods in solving linear equations.