Problem 46 - Intermediate Algebra

Question

Use Cramer's rule to solve each system of equations. $$ \left\\{\begin{array}{l} 5 x+3 y=72 \\ 3 x+5 y=56 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 12\) and \(y = 4\).

Step 1: Identify Coefficients and Constants

Start by identifying the coefficients of the variables and the constants from the system of equations. The system is: \( 5x + 3y = 72 \) and \( 3x + 5y = 56 \). The coefficients for \(x\) are \(5\) and \(3\); for \(y\) are \(3\) and \(5\). The constants on the right are \(72\) and \(56\).

Step 2: Determine the Main Determinant \(D\)

The determinant \(D\) of the coefficient matrix \(\begin{bmatrix} 5 & 3 \ 3 & 5 \end{bmatrix}\) is calculated as follows: \(D = (5)(5) - (3)(3) = 25 - 9 = 16\).

Step 3: Determine \(D_x\) for the Numerator of \(x\)

Replace the first column of the coefficient matrix with the constants to get the matrix \(\begin{bmatrix} 72 & 3 \ 56 & 5 \end{bmatrix}\). Calculate its determinant: \(D_x = (72)(5) - (3)(56) = 360 - 168 = 192\).

Step 4: Determine \(D_y\) for the Numerator of \(y\)

Replace the second column of the coefficient matrix with the constants to get the matrix \(\begin{bmatrix} 5 & 72 \ 3 & 56 \end{bmatrix}\). Calculate its determinant: \(D_y = (5)(56) - (72)(3) = 280 - 216 = 64\).

Step 5: Solve for \(x\) and \(y\)

Use Cramer's Rule to find \(x\) and \(y\). For \(x\), \(x = \frac{D_x}{D} = \frac{192}{16} = 12\). For \(y\), \(y = \frac{D_y}{D} = \frac{64}{16} = 4\).

Key Concepts

System of Equations

In mathematics, a system of equations is a collection of two or more equations with the same set of unknowns. The goal of solving such systems is to find the values of the unknowns that satisfy all equations simultaneously. In the given example, we have a system of two linear equations:

• \(5x + 3y = 72\)
• \(3x + 5y = 56\)

Each equation represents a line on a plane. The solution to this system is a point where both lines intersect, meaning it meets both conditions provided by the equations. Solving systems of equations can be accomplished using various methods, such as substitution, elimination, or matrix-based approaches like Cramer's rule, which is particularly useful for small 2x2 or 3x3 systems.

Determinants

Determinants are a special number that can be computed from a square matrix. They play a crucial role in linear algebra, particularly in solving linear systems using matrix techniques. In simple terms, the determinant tells us about important properties of a matrix, such as whether it has an inverse.For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated by the formula:\[D = ad - bc\]In the context of Cramer's rule, we compute the determinant of the coefficient matrix to understand if a unique solution exists. If the determinant is zero, the system may have no solution or infinitely many solutions. However, when it is non-zero, as in our example where \(D = 16\), it ensures a unique solution exists for the system of equations. This is a vital first step before applying Cramer's rule to solve for the unknowns.

Linear Algebra

Linear algebra is a field of mathematics dealing with vectors, vector spaces, linear transformations, and systems of linear equations. The principles of linear algebra are fundamental to diverse areas, including engineering, physics, computer science, and statistics. The core elements of linear algebra, such as matrices and determinants, provide a framework for systematically solving systems of linear equations. They allow for efficient computation and solving multiple variables simultaneously. Matrices act as an organizational tool for coefficients in systems of equations, while determinants indicate whether solutions exist. Linear algebra simplifies complex problem-solving processes. Cramer's rule, derived from linear algebra, particularly highlights this by providing a structured approach to find solutions with matrices, making complicated systems much simpler to handle and solve.

Solving Linear Systems

Solving linear systems is a fundamental part of mathematics and is a process that seeks the values of variables that satisfy a series of linear equations. Cramer's rule is one of the elegant methods within linear algebra for finding solutions of systems of linear equations, especially when the system has the same number of equations as unknowns. To use Cramer's rule, perform the following:

• First, construct the coefficient matrix from the system of linear equations.
• Calculate the determinant of this matrix. If it is non-zero, proceed; otherwise, the method cannot be applied.
• For each variable, replace the corresponding column in the coefficient matrix with the constant terms and calculate the determinant of this new matrix.
• The value of the variable is the determinant of this new matrix divided by the determinant of the original coefficient matrix.

For our given system, following Cramer's rule resulted in \(x=12\) and \(y=4\). Solving linear systems this way offers a clear path to the solution, provided all prerequisites like non-zero determinants are met.