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Magazine Chapter 13: Problem 60
Use Cramer's Rule to solve each system of equations. $$ \begin{array}{l}{3 x-4 y=13} \\ {-2 x+5 y=-4}\end{array} $$
Short Answer
Expert verified
The solution is \(x = 7\) and \(y = 2\).
Step by step solution
01
Identify the Coefficients Matrix
First, identify the matrix of coefficients from the system of equations. The system is given as: \[\begin{align*} 3x - 4y &= 13, \ -2x + 5y &= -4. \end{align*}\] The coefficients matrix \(A\) is: \[A = \begin{bmatrix} 3 & -4 \ -2 & 5 \end{bmatrix}.\]
02
Calculate the Determinant of the Coefficients Matrix
Find the determinant of matrix \(A\). The formula for the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(ad - bc\). Here, \(a = 3\), \(b = -4\), \(c = -2\), and \(d = 5\). Thus, the determinant \(\det(A) = 3 \cdot 5 - (-4) \cdot (-2) = 15 - 8 = 7.\)
03
Form the Matrices for Variables
Create matrices \(A_x\) for \(x\) and \(A_y\) for \(y\) by replacing the respective column of the matrix \(A\) with the constants from the equations. For \(x\), replace the first column with the constants: \[A_x = \begin{bmatrix} 13 & -4 \ -4 & 5 \end{bmatrix}.\] For \(y\), replace the second column: \[A_y = \begin{bmatrix} 3 & 13 \ -2 & -4 \end{bmatrix}.\]
04
Calculate Determinants of New Matrices
Calculate \(\det(A_x)\) and \(\det(A_y)\). For \(A_x\): \[\det(A_x) = 13 \cdot 5 - (-4) \cdot (-4) = 65 - 16 = 49.\] For \(A_y\): \[\det(A_y) = 3 \cdot (-4) - 13 \cdot (-2) = -12 + 26 = 14.\]
05
Apply Cramer's Rule to Find Solutions
By Cramer's Rule, the solutions \(x\) and \(y\) are given by \(x = \frac{\det(A_x)}{\det(A)}\) and \(y = \frac{\det(A_y)}{\det(A)}\). Hence, \[x = \frac{49}{7} = 7,\] \[y = \frac{14}{7} = 2.\]
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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 consists of multiple equations working together. These equations share two or more variables. In our example, we have two linear equations: \\(3x - 4y = 13\) and \\(-2x + 5y = -4\). Here, both equations have the variables \(x\) and \(y\).
The goal when working with a system of equations is to find values for these variables that satisfy all equations simultaneously.
To solve them, various methods, like substitution, elimination, and Cramer's Rule, may be employed, depending on the scenario and structure of the equations.
The goal when working with a system of equations is to find values for these variables that satisfy all equations simultaneously.
To solve them, various methods, like substitution, elimination, and Cramer's Rule, may be employed, depending on the scenario and structure of the equations.
Determinant
The determinant is a special number you can calculate from a square matrix. It provides important information about the matrix, such as whether it has an inverse. In the case of a 2x2 matrix, the formula for the determinant is straightforward: \(det(A) = ad - bc\), where the matrix \(A\) looks like this:\[\begin{bmatrix} a & b \ c & d \end{bmatrix} \]The determinant helps in determining solutions for systems of equations using Cramer's Rule.
For our example where the coefficient matrix \(A\) was:\[\begin{bmatrix} 3 & -4 \ -2 & 5 \end{bmatrix} \]we calculated \(det(A) = 15 - 8 = 7\). This value is key to find the solutions of the system.
For our example where the coefficient matrix \(A\) was:\[\begin{bmatrix} 3 & -4 \ -2 & 5 \end{bmatrix} \]we calculated \(det(A) = 15 - 8 = 7\). This value is key to find the solutions of the system.
Matrices
Matrices are rectangular arrays of numbers. They are organized in rows and columns and can represent systems of equations, transformations, and more.
In the context of systems of equations, matrices help organize coefficients and constants neatly, allowing for systematic calculation methods such as Cramer's Rule.
For example, from the system \(3x - 4y = 13\) and \(-2x + 5y = -4\), we formed the coefficient matrix \(A\) as:\[\begin{bmatrix} 3 & -4 \ -2 & 5 \end{bmatrix} \]To apply Cramer's Rule, we create separate matrices for each variable: one by substituting the column with constants.
In the context of systems of equations, matrices help organize coefficients and constants neatly, allowing for systematic calculation methods such as Cramer's Rule.
For example, from the system \(3x - 4y = 13\) and \(-2x + 5y = -4\), we formed the coefficient matrix \(A\) as:\[\begin{bmatrix} 3 & -4 \ -2 & 5 \end{bmatrix} \]To apply Cramer's Rule, we create separate matrices for each variable: one by substituting the column with constants.
2x2 Matrix
A 2x2 matrix is a simple square matrix with two rows and two columns. It is particularly manageable, making it ideal for many introductory problems, such as solving linear systems with two equations using Cramer's Rule.
The simplicity allows straightforward computation of the determinant, which is essential for Cramer's Rule. For any 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), calculating the determinant is as simple as \(ad - bc\).
In solving our system of equations, the 2x2 setup facilitated the calculations for determinants and eventual solutions for \(x\) and \(y\), by straightforward substitution and arithmetic.
The simplicity allows straightforward computation of the determinant, which is essential for Cramer's Rule. For any 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), calculating the determinant is as simple as \(ad - bc\).
In solving our system of equations, the 2x2 setup facilitated the calculations for determinants and eventual solutions for \(x\) and \(y\), by straightforward substitution and arithmetic.
