Complete the proof of Cramer's Rule for two equations...

Complete the proof of Cramer's Rule for two equations containing two variables. [Hint: In system (5), page $586,$ if $a=0,$ then $b \neq 0$ and $c \neq 0,$ since $D=-b c \neq 0 .$ Now show that equation (6) provides a solution of the system when $a=0$. Then three cases remain: $b=0, c=0,$ and $d=0 .]$

Complete the proof of Cramer's Rule for two equations containing two variables.
[Hint: In system (5), page $586,$ if $a=0,$ then $b \neq 0$ and $c \neq 0,$ since $D=-b c \neq 0 .$ Now show that equation (6) provides a solution of the system when $a=0$. Then three cases remain: $b=0, c=0,$ and $d=0 .]$

Key Concepts

Systems of Linear Equations

This concept involves working with multiple equations that contain the same set of unknowns. In the context of two equations in two variables, solving the system means finding the unique values for the variables that satisfy both equations simultaneously. This subject is foundational in linear algebra as it introduces methodologies for solving for unknowns and understanding the interplay between different equations in a system.

Determinants

A determinant is a scalar value derived from a square matrix that provides key insights into the properties of the matrix, such as whether the matrix is invertible. In the case of a 2x2 matrix, the determinant is calculated as ad - bc. A non-zero determinant signals that the matrix is invertible, which is essential for the application of Cramer’s Rule in solving systems of linear equations.

Cramer’s Rule

Cramer’s Rule is a theorem in linear algebra that offers an explicit formula for the solution of a system of linear equations with as many equations as unknowns, given that the determinant of the coefficient matrix is non-zero. For a 2x2 system, it expresses each variable as a ratio of determinants, where the numerator is derived from replacing the corresponding column by the constant terms, and the denominator is the determinant of the coefficient matrix.

Conditions for a Unique Solution

A unique solution for a system of linear equations is guaranteed when the determinant of the coefficient matrix is non-zero. This condition ensures that the system is non-singular and that each variable can be isolated uniquely using techniques like Cramer’s Rule. The necessity of a non-zero determinant underpins the entire proof of Cramer’s Rule and is also crucial when handling special cases in the proof.

Case Analysis in Proofs

Case analysis is an essential method in mathematical proofs that involves breaking down the proof into several distinct cases that cover all potential situations. In the proof of Cramer’s Rule, special attention is given to cases where specific coefficients (such as a, b, c, or d) are zero, ensuring that the theorem holds true in every scenario and that the formula derived continues to provide accurate solutions. This rigorous handling of cases solidifies the overall proof even when certain parameters assume zero values.

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