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Chapter 12: Problem 22

Solve each system of equations using Cramer's Rule if is applicable. If Cramer's Rule is not applicable, write, "Not applicable" \(\left\\{\begin{array}{l}2 x+4 y=16 \\ 3 x-5 y=-9\end{array}\right.\)

Short Answer

Expert verified
The solutions are \(x = 2\) and \(y = 3\).

Step by step solution

01

Write the coefficient matrix

Identify and write the coefficients of the variables from each equation in matrix form. The coefficient matrix for the system given is: \[ A = \begin{pmatrix} 2 & 4 \ 3 & -5 \end{pmatrix} \]
02

Compute the determinant of the coefficient matrix

To determine if Cramer's Rule is applicable, compute the determinant of the coefficient matrix. \[\text{det}(A) = \begin{vmatrix} 2 & 4 \ 3 & -5 \end{vmatrix} = (2 \times -5) - (4 \times 3) = -10 - 12 = -22 \]
03

Check if the determinant is zero

Since \(\text{det}(A) = -22 eq 0\), Cramer's Rule is applicable, and we can proceed to find the solutions for \(x\) and \(y\).
04

Compute the determinant for x

Replace the first column of the coefficient matrix with the constants from the right-hand side of the equations and find the determinant. \[ A_x = \begin{pmatrix} 16 & 4 \ -9 & -5 \end{pmatrix} \] \[\text{det}(A_x) = \begin{vmatrix} 16 & 4 \ -9 & -5 \end{vmatrix} = (16 \times -5) - (4 \times -9) = -80 + 36 = -44 \]
05

Compute the determinant for y

Replace the second column of the coefficient matrix with the constants from the right-hand side of the equations and find the determinant. \[ A_y = \begin{pmatrix} 2 & 16 \ 3 & -9 \end{pmatrix} \] \[\text{det}(A_y) = \begin{vmatrix} 2 & 16 \ 3 & -9 \end{vmatrix} = (2 \times -9) - (16 \times 3) = -18 - 48 = -66 \]
06

Find the values of x and y

Use Cramer's Rule to find the values of \(x\) and \(y\): \[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-44}{-22} = 2 \] \[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{-66}{-22} = 3 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Equations
A system of equations is a collection of two or more equations with the same set of variables. In our case, we are dealing with two equations involving two variables, x and y. A system of equations can be solved using various methods such as substitution, elimination, or matrix-based methods like Cramer's Rule. Understanding how to translate a word problem into a system of equations is crucial in solving real-world problems. By aligning the equations in a standard form鈥攅ach variable in a column, constants on the other side鈥攚e create a clear path toward a solution.
Determinant
In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix. It helps to solve systems of linear equations, among other applications. The determinant provides important properties such as whether a matrix is invertible. To compute the determinant of a 2x2 matrix like \(\begin{pmatrix}a & b \ c & d\text \)\begin{pmatrix}\text), we use the formula \(\text{det}(A) = ad - bc\text \). This computation is foundational for applying Cramer's Rule, as seen in the given exercise.
Matrix Algebra
Matrix algebra involves operations with matrices, including addition, subtraction, multiplication, and finding inverses. Understanding these operations is essential when solving linear equations using methods such as Cramer's Rule.
In our example, we start by forming the coefficient matrix:\(\begin{pmatrix}2 & 4 \ 3 & -5\text \)\begin{pmatrix}\text). Matrix algebra helps to systematically handle and manipulate the data stored in these matrices, leading to efficient solutions.
Linear Algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but also extends to more abstract and higher-dimensional spaces. Linear algebra plays a significant role in diverse areas such as computer science, engineering, physics, and economics. Specific to our exercise, concepts such as the determinant and matrix operations are core tools from linear algebra used to solve systems of linear equations.

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