Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{l} 2 x+5 y=16 \\ 3 x-7 y=24 \end{array}\right. $$

Step-by-Step Solution

Answer

x = 8, y = 0.

Step 1: Identify the Coefficient Matrix

The given system of equations is: \(2x + 5y = 16\) and \(3x - 7y = 24\). The coefficient matrix, \(A\), is formed using the coefficients of \(x\) and \(y\). Thus, \(A = \begin{pmatrix} 2 & 5 \ 3 & -7 \end{pmatrix}\).

Step 2: Compute the Determinant of the Coefficient Matrix

Calculate the determinant of \(A\) using \(\det(A) = (2)(-7) - (5)(3)\). This simplifies to \(-14 - 15 = -29\). The determinant \(\det(A) = -29\) is nonzero, so Cramer's rule is applicable.

Step 3: Formulate the Matrix for Variable x

To find \(x\), replace the first column of \(A\) with the constants from the right side of the equations to form matrix \(A_x\). \(A_x = \begin{pmatrix} 16 & 5 \ 24 & -7 \end{pmatrix}\).

Step 4: Compute the Determinant of Matrix A_x

Calculate the determinant of \(A_x\) using \(\det(A_x) = (16)(-7) - (5)(24)\). This simplifies to \(-112 - 120 = -232\).

Step 5: Solve for x

Using Cramer’s Rule, \(x = \frac{\det(A_x)}{\det(A)} = \frac{-232}{-29}\). Simplifying gives \(x = 8\).

Step 6: Formulate the Matrix for Variable y

To find \(y\), replace the second column of \(A\) with the constants from the right side of the equations to form matrix \(A_y\). \(A_y = \begin{pmatrix} 2 & 16 \ 3 & 24 \end{pmatrix}\).

Step 7: Compute the Determinant of Matrix A_y

Calculate the determinant of \(A_y\) using \(\det(A_y) = (2)(24) - (16)(3)\). This simplifies to \(48 - 48 = 0\).

Step 8: Solve for y

Using Cramer’s Rule, \(y = \frac{\det(A_y)}{\det(A)} = \frac{0}{-29}\). This simplifies to \(y = 0\).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that represent straight lines. They consist of variables with constant coefficients and no higher powers, such as squares or cubes. In our system, the equations are \( 2x + 5y = 16 \) and \( 3x - 7y = 24 \). Here is what defines a linear equation:

• Variables: They represent unknown quantities, often written as \( x \), \( y \), etc.
• Coefficients: These are the numbers multiplying the variables, here 2, 5, 3, and -7.
• Constants: These are the terms without variables, like 16 and 24.

This system of equations is considered consistent and can be solved simultaneously using various methods, such as Cramer's rule. They form a foundational building block for more complicated algebraic expressions.

Determinant

The determinant is a special number that can be calculated from a square matrix. It provides crucial information about the matrix; for instance, whether the matrix is invertible. For a 2x2 matrix like our coefficient matrix \( A \), it gives us an easy way to determine the matrix's properties. To find it, use the formula: \[ \det(A) = ad - bc = (2)(-7) - (5)(3) = -14 - 15 = -29 \]

• If the determinant is nonzero, the matrix has an inverse, and for systems of equations, it implies a unique solution.
• If the determinant is zero, it means the matrix does not have an inverse, and there might be no solution or infinitely many solutions.

Thus, since \( \det(A) = -29 \), Cramer's Rule can be applied to this system.

Coefficient Matrix

A coefficient matrix is composed solely of the coefficients from the variables of a system of linear equations. For our problem, the coefficient matrix \( A \) is \[ A = \begin{pmatrix} 2 & 5 \ 3 & -7 \end{pmatrix} \] This matrix allows us to systematically solve the equations using various methods like Cramer's Rule or matrix algebra. Here's a breakdown of the roles within the matrix:

• Each row represents an equation.
• Each column represents a different variable’s coefficients.

Understanding how to construct and manipulate a coefficient matrix is essential for leveraging matrix-based solving techniques. In our case, it helped in determining the solvability and finding the solution of the system of equations.

Matrix Algebra

Matrix algebra involves performing operations with matrices to solve equations or transform data. It is an extension of algebraic concepts to matrix manipulation. Some key operations involve addition, subtraction, multiplication, and finding determinants, as we did with matrices \( A \), \( A_x \), and \( A_y \):

• Transformation: Replacing columns with constant terms to form \( A_x \) or \( A_y \).
• Determinant Calculation: Used to apply Cramer's Rule.

In our exercise, matrix algebra was pivotal in implementing Cramer's Rule. It allowed us to manipulate matrices to extract values for the variables \( x \) and \( y \) using transformations and determinant calculations as shown. Understanding these concepts is crucial for a deeper comprehension of systems of linear equations and their solutions.