Problem 21 - College Algebra

Question

Use Cramer's Rule to solve each system. $$\left\\{\begin{array}{l}2 x=3 y+2 \\\5 x=51-4 y\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 6.826\) and \(y = 4\).

Step 1: Write the system in standard form

First, we can start by writing the given equations in standard form: \(2x - 3y =2\) , \(5x + 4y = 51\).

Step 2: Form the coefficient matrix and calculate its determinant

Next, form the matrix of coefficients of the unknowns, \(x\) and \(y\), and calculate its determinant: \[\left(\begin{array}{cc}2 & -3 \ 5 & 4 \end{array}\right)\]. The determinant, denoted as D, is calculated as: \(D = (2*4) - (-3*5) = 23\).

Step 3: Form matrices Dx and Dy and calculate their determinants

Form a matrix \(D_x\) by replacing the x-coefficients in D with the constants and calculate its determinant: \[\left(\begin{array}{cc}2 & -3 \ 51 & 4 \end{array}\right)\]. The determinant \(D_x\) is equal to: \(D_x = (2*4) - (-3*51) = 157\). Similarly, form \(D_y\) by replacing the y-coefficients in D with the constants and calculate its determinant: \[\left(\begin{array}{cc}2 & 2 \ 5 & 51 \end{array}\right)\]. Here, \(D_y = (2*51) - (5*2) = 92\).

Step 4: Solve for x and y using Cramer's rule

Lastly, we use the Cramer's rule formula to calculate the values of \(x\) and \(y\). \(x = D_x / D = 157 / 23 = 6.826\), and \(y = D_y / D = 92 / 23 = 4\).

Key Concepts

System of Equations

A system of equations consists of multiple equations that can have two or more unknowns to solve for. In this case, our system is composed of two linear equations with unknowns for variables \(x\) and \(y\). We aim to find the values of \(x\) and \(y\) that satisfy both equations. By writing these equations in a standard format, they can easily be analyzed and solved using various methods.

• Equations: \(2x - 3y = 2\) and \(5x + 4y = 51\)
• Unknowns: \(x\) and \(y\)
• Standard Form: Align the equations so they display general forms: \(ax + by = c\)

By clearly structuring the equations, we facilitate the use of solutions techniques like substitution, elimination, or in this case, Cramer's Rule.

Determinant

The determinant is a special number that can be computed from a square matrix, and it plays a crucial role in solving systems of equations using Cramer's Rule. For our 2x2 coefficient matrix, the determinant helps determine if a unique solution exists.

• The matrix of coefficients: \(\begin{bmatrix} 2 & -3 \ 5 & 4 \end{bmatrix}\)
• Formula: For a 2x2 matrix, the determinant \(D\) is given by \(ad - bc\), where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix.
• Calculation: \(D = (2)(4) - (-3)(5) = 23\)

Since the determinant is non-zero (23), we know that a unique solution exists for the system using Cramer's Rule.

Matrix

A matrix is an array of numbers arranged in rows and columns. In the context of solving equations, a matrix can be used to organize and simplify calculations. For Cramer's Rule, we work primarily with the coefficient matrix and its variants.

• Coefficient Matrix: Contains the coefficients of the unknowns from the system of equations.
• \(D\): \(\begin{bmatrix} 2 & -3 \ 5 & 4 \end{bmatrix}\)
• Transformation Matrices (\(D_x, D_y\)): Created by substituting columns from the coefficient matrix with the constants from the equations.

Transforming the coefficient matrix into \(D_x\) and \(D_y\) allows us to apply Cramer's Rule by generating specific determinants used to solve for each variable individually.

Algebra

Algebra provides the framework for manipulating equations and finding variable values. By employing algebraic principles, methods like Cramer's Rule translate to practical solutions for systems of equations.

• Cramer's Rule: An algebraic method for solving linear systems where the number of equations matches the number of unknowns, using determinants.
• Formula: \(x = \frac{D_x}{D}\), \(y = \frac{D_y}{D}\)
• Calculation: For our problem, \(D_x = 157\) and \(D_y = 92\), thus \(x = \frac{157}{23}\) and \(y = \frac{92}{23}\).

This rule highlights how algebra streamlines the solution process, breaking it down into clear steps with precise calculations to find each unknown.