Problem 33 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Coefficients

We start by identifying the coefficients of the variables in the given system of equations. Let's write them down:1. The coefficients of the first equation \(4x + 10y = 180\) are 4 and 10.2. The coefficients of the second equation \(-3x - 5y = -105\) are -3 and -5.

Step 2: Form the Coefficient Matrix

Using the coefficients from the equations, we form the coefficient matrix: \[ A = \begin{pmatrix} 4 & 10 \ -3 & -5 \end{pmatrix} \]

Step 3: Calculate the Determinant of the Coefficient Matrix

Find the determinant of the matrix \(A\) using the formula for 2x2 matrices: \( \text{det}(A) = (4)(-5) - (10)(-3) = -20 + 30 = 10 \)

Step 4: Form the Matrices for Numerators

Next, form the matrices for numerators by replacing one column at a time with the constants from the equations.For \(x\), replace the first column of \(A\) with the constants:\[ A_x = \begin{pmatrix} 180 & 10 \ -105 & -5 \end{pmatrix} \]For \(y\), replace the second column of \(A\) with the constants:\[ A_y = \begin{pmatrix} 4 & 180 \ -3 & -105 \end{pmatrix} \]

Step 5: Calculate Determinants for Numerator Matrices

Calculate the determinant for each of the matrices.\( \text{det}(A_x) = (180)(-5) - (10)(-105) = -900 - (-1050) = 150 \)\( \text{det}(A_y) = (4)(-105) - (180)(-3) = -420 + 540 = 120 \)

Step 6: Apply Cramer's Rule

Use Cramer's Rule to find the solutions for \(x\) and \(y\):For \(x\):\( x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{150}{10} = 15 \)For \(y\):\( y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{120}{10} = 12 \)

Key Concepts

System of Linear Equations

A system of linear equations comprises two or more linear equations involving the same set of variables. In this exercise, we have two equations with two variables, \(x\) and \(y\):

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

These equations must be solved simultaneously to find values of \(x\) and \(y\) that satisfy both equations. The goal of solving a system is to find the point where these lines intersect on a graph.
The process of solving them involves finding the values of the variables where all the equations hold true.
Methods to solve them include substitution, elimination, and the focus here: Cramer's Rule.

Determinant Calculation

The determinant is a scalar value calculated from a square matrix. It provides important properties of a matrix and helps in solving systems of linear equations using Cramer's Rule. To find the determinant for a 2x2 matrix \(A\), use the formula:\[\text{det}(A) = ad - bc\]where \(A\) is of the form:\[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]
In our exercise, the coefficient matrix \(A\) is \[\begin{pmatrix} 4 & 10 \ -3 & -5 \end{pmatrix}\] Plugging in the values,\[\text{det}(A) = (4)(-5) - (10)(-3) = -20 + 30 = 10\]Calculating the determinant is crucial because for Cramer's Rule to apply, the determinant must be non-zero. A zero determinant indicates that the equations do not have a unique solution.

Coefficient Matrix

The coefficient matrix is a matrix formed from the coefficients of the variables in the system of linear equations. For our exercise, the coefficient matrix is made from the variables in the equations \(4x + 10y = 180\) and \(-3x - 5y = -105\).
This matrix is represented as:\[A = \begin{pmatrix} 4 & 10 \ -3 & -5 \end{pmatrix}\] This matrix helps in organizing the system of equations and is used to calculate the determinant, as discussed earlier.
The ability to easily visualize and handle the coefficients makes matrices a powerful tool in linear algebra and solving systems of equations.

Numerator Matrices

Numerator matrices play a key role in Cramer's Rule. They are derived from the coefficient matrix by substituting one column at a time with the constants from the right-hand side of the equations.
When solving for \(x\), you replace the first column:\[A_x = \begin{pmatrix} 180 & 10 \ -105 & -5 \end{pmatrix}\] For \(y\), substitute the second column:\[A_y = \begin{pmatrix} 4 & 180 \ -3 & -105 \end{pmatrix}\]
Calculating the determinant for each numerator matrix provides the values needed for Cramer's Rule. Specifically:- \(\text{det}(A_x) = 150\) - \(\text{det}(A_y) = 120\)
The process illustrates how changes in the coefficients influence the outcome, leading to the unique solution for each variable through Cramer's Rule.