Problem 74 - Algebra and Trigonometry

Question

Consider the system $$ \left\\{\begin{array}{l} {a_{1} x+b_{1} y=c_{1}} \\ {a_{2} x+b_{2} y=c_{2}} \end{array}\right. $$ 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

Upon simplifying the determinants in Cramer's Rule for both the original system of equations and the new system with the first equation replaced by the sum of both equations, it can be proved mathematically that the solutions \(x = x'\) and \(y = y'\), therefore, confirming the solutions are indeed the same.

Step 1: State the system

The original system of linear equations can be stated as: \[ a_{1} x+b_{1} y = c_{1} \] \[ a_{2} x+b_{2} y = c_{2} \]

Step 2: Convert into matrix form

This system can be represented as the following matrix equation: \[ \begin{bmatrix} a_{1} & b_{1} \ a_{2} & b_{2} \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} c_{1} \ c_{2} \end{bmatrix} \]

Step 3: Apply Cramer's Rule and find values of X and Y

Using Cramer's Rule, the solution, \((x, y)\), can be found by calculating the determinants of matrices. \[ x = \frac{\begin{vmatrix} c_{1} & b_{1} \ c_{2} & b_{2} \end{vmatrix}}{\begin{vmatrix} a_{1} & b_{1} \ a_{2} & b_{2} \end{vmatrix}}\] \[ y = \frac{\begin{vmatrix} a_{1} & c_{1} \ a_{2} & c_{2} \end{vmatrix}}{\begin{vmatrix} a_{1} & b_{1} \ a_{2} & b_{2} \end{vmatrix}}\]

Step 4: Sum of the two equations

Now, replace the first equation with the sum of both the equations, the system becomes \[ (a_{1} + a_{2})x + (b_{1} + b_{2})y = c_{1} + c_{2} \] \[ a_{2}x + b_{2}y = c_{2} \]

Step 5: Apply Cramer's Rule on the new system

From the new system, apply the Cramer's Rule again to find the solution as follows: \[ x' = \frac{\begin{vmatrix} c_{1} + c_{2} & b_{1} + b_{2} \ c_{2} & b_{2} \end{vmatrix}}{\begin{vmatrix} a_{1}+a_{2} & b_{1}+b_{2} \ a_{2} & b_{2} \end{vmatrix}}\] \[ y' = \frac{\begin{vmatrix} a_{1}+a_{2} & c_{1}+c_{2} \ a_{2} & c_{2} \end{vmatrix}}{\begin{vmatrix} a_{1}+a_{2} & b_{1}+b_{2} \ a_{2} & b_{2} \end{vmatrix}}\]

Step 6: Compare the solutions

By simplifying the determinants, it can be proved that \(x = x'\) and \(y = y'\), which means the solutions to the original system and the new system are the same.

Key Concepts

Linear Equations

Linear equations are the building blocks of algebra and form the essence of mathematical modeling in various fields. These equations represent straight lines when graphed on a coordinate plane. In its simplest form, a linear equation in two variables can be expressed as \( ax + by = c \). Here, \( a \) and \( b \) are coefficients representing the slope of the line, \( x \) and \( y \) are variables, and \( c \) is a constant.

Linear equations are defined by the characteristic that the highest power of the variable is 1, making the graph a straight line. This foundational concept allows us to model relationships where change is constant, meaning for every change in \( x \), \( y \) changes proportionally.

Understanding linear equations is crucial, as they are essential for solving more complex topics like systems of equations and applying methods like Cramer's Rule.

Determinants

Determinants are key mathematical tools used in solving systems of linear equations. They are scalars that can be computed from a square matrix and provide significant information about the matrix, such as whether it is invertible. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \).

This numerical value helps decide many properties of a matrix. Specifically, if the determinant is zero, the system of equations has no unique solution or infinitely many solutions. This characteristic becomes especially relevant when using Cramer's Rule to solve systems of equations.

Cramer's Rule relies heavily on determinants to find the solution of each variable in a system, ensuring each solution is valid only when the determinant of the coefficient matrix is non-zero.

Systems of Equations

Systems of equations are sets of equations with multiple variables that share solutions. They appear frequently in mathematics to represent situations where multiple conditions must be satisfied simultaneously. A simple example is solving for two unknown values, \( x \) and \( y \), in two linear equations. For example, \( a_1 x + b_1 y = c_1 \) and \( a_2 x + b_2 y = c_2 \).

There are three potential outcomes when solving systems of equations: exactly one solution, no solution, or infinitely many solutions. The nature of the solutions largely depends on the relationship between the equations.

Cramer's Rule is a method used for tackling systems of equations, primarily when the number of equations equals the number of unknowns. By using determinants of matrices crafted from the coefficients of these equations, Cramer's Rule systematically isolates each variable to find its solution efficiently. This method is efficient for computational solutions, especially when the system has a unique solution.