Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{r} 4 x-5 y=7 \\ -3 x+9 y=0 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = 3\) and \(y = 1\).

Step 1: Write Down the System of Equations

Consider the system of linear equations given: \( 4x - 5y = 7 \) and \( -3x + 9y = 0 \). These equations are in the standard form of \( ax + by = c \).

Step 2: Set Up the Coefficient Matrix

Write the coefficient matrix based on the coefficients of \(x\) and \(y\) in the equations:\[A = \begin{bmatrix} 4 & -5 \ -3 & 9 \end{bmatrix}\]

Step 3: Calculate the Determinant of the Coefficient Matrix

Use the formula for the determinant of a 2x2 matrix, \( \text{det}(A) = ad - bc \), to find the determinant:\[\text{det}(A) = (4)(9) - (-5)(-3) = 36 - 15 = 21\]

Step 4: Set Up the Matrix for Solving x

To solve for \(x\), replace the first column of \(A\) with the constants from the right-hand side of the equations:\[A_x = \begin{bmatrix} 7 & -5 \ 0 & 9 \end{bmatrix}\]

Step 5: Calculate the Determinant of Matrix A_x

Find the determinant of the matrix \(A_x\):\[\text{det}(A_x) = (7)(9) - (-5)(0) = 63 - 0 = 63\]

Step 6: Solve for x using Cramer's Rule

Apply Cramer's Rule to solve for \(x\): \[x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{63}{21} = 3\]

Step 7: Set Up the Matrix for Solving y

To solve for \(y\), replace the second column of \(A\) with the constants from the equations:\[A_y = \begin{bmatrix} 4 & 7 \ -3 & 0 \end{bmatrix}\]

Step 8: Calculate the Determinant of Matrix A_y

Find the determinant of the matrix \(A_y\):\[\text{det}(A_y) = (4)(0) - (7)(-3) = 0 + 21 = 21\]

Step 9: Solve for y using Cramer's Rule

Apply Cramer's Rule to solve for \(y\): \[y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{21}{21} = 1\]

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. These equations can be represented in the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. The goal when solving a system of linear equations is to find the values of the variables that satisfy all equations simultaneously.

Linear equations can occur in various real-world scenarios, such as calculating finances, predicting population growth, and more. They are called linear because their graphs are straight lines.

• Each equation in the system can be represented graphically as a line on the coordinate plane.
• The solution is the point where all the lines intersect.

In the given exercise, we have a system of two linear equations involving two variables, \(x\) and \(y\):\[\begin{align*}4x - 5y &= 7 \-3x + 9y &= 0\end{align*}\]Solving this system using Cramer's Rule involves working with matrices and determinants which we will explain further.

Determinant

The determinant is a special number that can be calculated from a square matrix. It provides important information about the matrix and the system of equations that the matrix represents. In the case of a 2x2 matrix, the determinant can be calculated using the formula:\[\text{det}(A) = ad - bc\]where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix:\[A = \begin{bmatrix}a & b \c & d\end{bmatrix}\]In our exercise, the coefficient matrix \(A\) is:\[A = \begin{bmatrix} 4 & -5 \-3 & 9 \end{bmatrix}\]The determinant is calculated as follows:\[\text{det}(A) = (4)(9) - (-5)(-3) = 36 - 15 = 21\]A non-zero determinant, like 21 in our case, indicates that the system of equations has a unique solution. This step is crucial in verifying that Cramer's Rule is applicable.

Coefficient Matrix

The coefficient matrix is constructed using the coefficients of the variables from each equation in the system. It forms the backbone of approaches like Cramer's Rule. This matrix isolates the numerical components of each equation without the variables or constants, which helps in conducting matrix operations like finding determinants.

For our given system:\[\begin{align*}4x - 5y &= 7 \-3x + 9y &= 0\end{align*}\]the coefficient matrix \(A\) is:\[A = \begin{bmatrix}4 & -5 \-3 & 9\end{bmatrix}\]This matrix highlights:

• The first row contains the coefficients of \(x\) and \(y\) from the first equation.
• The second row contains the coefficients from the second equation.

By setting up the coefficient matrix, we can systematically apply Cramer's Rule and other mathematical techniques to solve the system of equations.

Algebra

Algebra provides the foundational techniques and tools for solving equations and systems of equations, making it integral to understanding methods like Cramer's Rule. It involves the manipulation of symbols and variables to solve problems.

In the context of this exercise, algebraic methods are used in arranging and simplifying the coefficients and constants to apply Cramer's Rule effectively. Here are some key points:

• Algebra helps derive the coefficient matrix by separating constants and variables.
• It aids in setting up and calculating determinants by organizing the matrix elements.
• Finally, algebraic division is used when applying Cramer's Rule, dividing the determinants to find the variable values.

Cramer's Rule is a sophisticated algebraic technique that allows solving systems by leveraging determinants. It exemplifies the power of algebra in encoding and unraveling complex mathematical relationships to find simple variable solutions.