Problem 45 - Algebra 2

Question

Use Cramer’s Rule to solve each system of equations. \(\frac{x}{2}-\frac{2 y}{3}=2 \frac{1}{3}\) \(3 x+4 y=-50\)

Step-by-Step Solution

Answer

The solution is \( x = -6 \) and \( y = -8 \).

Step 1: Rewrite Equations in Standard Form

First, convert the equations into standard form. The standard form for this system is: 1. \( \frac{x}{2} - \frac{2y}{3} = \frac{7}{3} \) 2. \( 3x + 4y = -50 \) Multiply the first equation by 6 to eliminate the fractions:\[ 3x - 4y = 14 \] Now we have:1. \( 3x - 4y = 14 \) 2. \( 3x + 4y = -50 \)

Step 2: Form the Coefficient Matrix

Extract the coefficients from the system to form the coefficient matrix \( A \):\[A = \begin{bmatrix} 3 & -4 \ 3 & 4 \end{bmatrix} \]

Step 3: Calculate the Determinant of the Coefficient Matrix

Calculate the determinant of matrix \( A \) using:\[ |A| = \begin{vmatrix} 3 & -4 \ 3 & 4 \end{vmatrix} = (3)(4) - (-4)(3) = 12 + 12 = 24 \]

Step 4: Form Matrix for x (Replace Column 1)

Replace the first column of \( A \) with the constants from the right side of the equations:\[A_x = \begin{bmatrix} 14 & -4 \ -50 & 4 \end{bmatrix}\]

Step 5: Calculate Determinant for x

Calculate the determinant of \( A_x \):\[|A_x| = \begin{vmatrix} 14 & -4 \ -50 & 4 \end{vmatrix} = (14)(4) - (-4)(-50) = 56 - 200 = -144\]

Step 6: Form Matrix for y (Replace Column 2)

Replace the second column of \( A \) with the constants from the right side:\[A_y = \begin{bmatrix} 3 & 14 \ 3 & -50 \end{bmatrix}\]

Step 7: Calculate Determinant for y

Calculate the determinant of \( A_y \):\[|A_y| = \begin{vmatrix} 3 & 14 \ 3 & -50 \end{vmatrix} = (3)(-50) - (14)(3) = -150 - 42 = -192\]

Step 8: Solve for x and y

Apply Cramer's Rule to solve for \( x \) and \( y \):\[x = \frac{|A_x|}{|A|} = \frac{-144}{24} = -6 \]\[y = \frac{|A_y|}{|A|} = \frac{-192}{24} = -8 \]

Key Concepts

Understanding Systems of Equations

A system of equations is a set of two or more equations that share two or more unknowns. The goal is to find values for these unknowns that satisfy each equation simultaneously. These systems can often be visualized as lines in a coordinate plane.
Their intersection points represent the solutions, where each equation in the system holds true. Typically, systems of equations can be expressed as:

• Linear equations (straight lines)
• Quadratic equations (parabolic curves)
• Other types of equations (circles, ellipses, etc.)

Our specific problem involved two linear equations. They needed to be transformed into a simpler and more workable format. This was accomplished by eliminating fractions, resulting in two convenient forms.
This step is crucial as it sets up the equations for methods like Cramer's Rule, used to find the solution easily.

Determinants and Their Role

The determinant is a special number calculated from the elements of a square matrix. Determinants are pivotal in matrix algebra and have various applications ranging from calculating areas to solving systems of equations like ours using Cramer's Rule.
The determinant can give insights into whether a matrix is invertible. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant \(|A|\) is worked out as \(ad - bc\).
In our problem, the determinant of the coefficient matrix is needed. It helps us assess whether there are unique solutions to the given system of equations. A non-zero determinant (as we calculated 24) implies that the system has a unique solution.
The determinants of matrices formed by modifying the original ones help us solve for each unknown independently. Non-zero values indicate valid solutions.

Diving into Matrix Algebra

Matrix algebra involves the study of matrices and operations that can be performed on them. Matrices are rectangular arrays of numbers, strings, or variables. They are crucial in many fields such as computer graphics, physics, and statistics.
In solving systems of equations, matrices serve as a compact and systematic way to organize data. This is especially true when dealing with multiple equations with unknowns.
Different types of matrices include:

• Square (equal number of rows and columns)
• Rectangular (different number of rows and columns)
• Diagonal or identity matrices

The use of matrices in our exercise helped to formulate the coefficient matrix. It supports the structured approach provided by Cramer’s Rule, making it easier to find the solution for \(x\) and \(y\).
By replacing columns in the coefficient matrix with constants from the equations, new matrices were formed to solve for each variable separately, utilizing determinants to find specific values.