Problem 33 - College Algebra

Question

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{c} 4 x+10 y=180 \\ -3 x-5 y=-105 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = 15\), \(y = 12\).

Step 1: Identify the Coefficient Matrix

First, let's identify the coefficients from the system of equations given: \(4x + 10y = 180\) and \(-3x - 5y = -105\). The coefficient matrix \(A\) is: \[ A = \begin{bmatrix} 4 & 10 \ -3 & -5 \end{bmatrix} \]

Step 2: Calculate Determinant of the Coefficient Matrix

Calculate the determinant of matrix \(A\). For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is found using the formula \(ad - bc\).Apply the formula:\[ \text{det}(A) = (4)(-5) - (10)(-3) \]\[ \text{det}(A) = -20 + 30 = 10 \]

Step 3: Formulate Matrices for Cramer's Rule

According to Cramer's Rule, we need two more matrices: one by replacing the first column of \(A\) with the constants (\(180\) and \(-105\)), the second by replacing the second column of \(A\) with the constants.\[ A_x = \begin{bmatrix} 180 & 10 \ -105 & -5 \end{bmatrix} \]\[ A_y = \begin{bmatrix} 4 & 180 \ -3 & -105 \end{bmatrix} \]

Step 4: Calculate Determinants of Modified Matrices

Calculate the determinants of \(A_x\) and \(A_y\).For \(A_x\):\[ \text{det}(A_x) = (180)(-5) - (10)(-105) \]\[ \text{det}(A_x) = -900 + 1050 = 150 \]For \(A_y\):\[ \text{det}(A_y) = (4)(-105) - (180)(-3) \]\[ \text{det}(A_y) = -420 + 540 = 120 \]

Step 5: Solve for x and y using Cramer's Rule

Use Cramer's Rule which states \(x = \frac{\text{det}(A_x)}{\text{det}(A)}\) and \(y = \frac{\text{det}(A_y)}{\text{det}(A)}\).For \(x\):\[ x = \frac{150}{10} = 15 \]For \(y\):\[ y = \frac{120}{10} = 12 \]

Step 6: Verify the solution

Substitute \(x = 15\) and \(y = 12\) back into the original equations to verify:For \(4x + 10y = 180\):\[ 4(15) + 10(12) = 60 + 120 = 180 \] - which is correct.For \(-3x - 5y = -105\):\[ -3(15) - 5(12) = -45 - 60 = -105 \] - which is also correct.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations that share common variables. In this case, we have two equations involving the variables \( x \) and \( y \). These are written as:

• First equation: \( 4x + 10y = 180 \)
• Second equation: \( -3x - 5y = -105 \)

This system is often solved to find values of \( x \) and \( y \) that satisfy both equations simultaneously. Such dual-equations can graphically be represented as straight lines on a coordinate plane. The solution corresponds to the point where these lines intersect.
The process of finding this intersection point algebraically can be simplified using a technique called Cramer's Rule, which utilizes determinants and matrices.

Determinant

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \text{det}(A) \) is calculated using the formula:\[ \text{det}(A) = ad - bc \]This scalar value plays a crucial role in assessing the properties of the matrix, such as whether the matrix is invertible, and it is central in Cramer's Rule for solving linear equations.
In the context of solving a system of equations, calculating the determinant of the coefficient matrix helps us find the unique solution through Cramer's Rule, as long as the determinant is non-zero.

Coefficient Matrix

The coefficient matrix is formed by taking the coefficients of variables from a system of linear equations and arranging them in a square matrix. For example, in our exercise, the coefficients from \( 4x + 10y = 180 \) and \( -3x - 5y = -105 \) are organized into a matrix:\[ A = \begin{bmatrix} 4 & 10 \ -3 & -5 \end{bmatrix} \]Each row in this matrix corresponds to an equation from the system, while each column corresponds to a variable's coefficients.

• First column represents coefficients of \( x \): \( 4 \) and \( -3 \).
• Second column represents coefficients of \( y \): \( 10 \) and \( -5 \).

The coefficient matrix is fundamental in using Cramer's Rule, as it allows us to compute necessary determinants for solving the system.

2x2 Matrix

A 2x2 matrix is a simple structure consisting of two rows and two columns. It is used to organize data and simplifies solving systems with two equations and two unknowns. In our problem, several 2x2 matrices are used:

• The original coefficient matrix \( A \).
• Modified matrices \( A_x \) and \( A_y \), where columns are replaced with constants for computing determinants.

These matrices are crucial because Cramer's Rule revolves around calculating their determinants to find \( x \) and \( y \).
The simplicity of a 2x2 matrix makes it a practical and powerful tool in linear algebra, especially for beginners learning matrix operations and applications such as solving linear systems.