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Chapter 10: Problem 40

Use Cramer's Rule to solve the system. $$\left\\{\begin{aligned} 5 x-3 y+z &=6 \\ 4 y-6 z &=22 \\ 7 x+10 y &=-13 \end{aligned}\right.$$

Short Answer

Expert verified
The solution is approximately \( x=2.04 \), \( y=-2.65 \), \( z=-3.553 \).

Step by step solution

01

Write the System in Matrix Form

First, express the system of equations as a matrix equation. The system of equations is: \[5x - 3y + z = 6,4y - 6z = 22,7x + 10y = -13\] The matrix form is \( A\mathbf{x} = \mathbf{b} \), where matrix \( A \) is the coefficient matrix, \( \mathbf{x} \) is the variable matrix, and \( \mathbf{b} \) is the constants matrix: \[A = \begin{bmatrix} 5 & -3 & 1 \ 0 & 4 & -6 \ 7 & 10 & 0 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 6 \ 22 \ -13 \end{bmatrix}\]
02

Calculate Determinant of Coefficient Matrix

Calculate the determinant of matrix \( A \). To find the determinant of a \( 3\times3 \) matrix, use: \[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\] For matrix \( A \):\[\det(A) = 5(4 \cdot 0 - (-6) \cdot 10) - (-3)(0 \cdot 0 - (0) \cdot 7) + 1(0 \cdot 4 - (0) \cdot 7)\] \[= 5(60) + 0 + 0 = 300\]
03

Calculate Determinant of Each Modified Matrix

Create matrices \( A_x, A_y, \) and \( A_z \) by replacing the respective column of \( A \) with the column vector \( \mathbf{b} \). Then, calculate their determinants.**Matrix \( A_x \):**\[ A_x = \begin{bmatrix} 6 & -3 & 1 \ 22 & 4 & -6 \ -13 & 10 & 0 \end{bmatrix} \] \[ \det(A_x) = 6(4 \cdot 0 - (-6) \cdot 10) - (-3)(22 \cdot 0 - (0) \cdot -13) + 1(22 \cdot 10 - 4 \cdot -13) \] \[ = 6(60) + 0 + (220 + 52) \] \[ = 612 \]**Matrix \( A_y \):**\[ A_y = \begin{bmatrix} 5 & 6 & 1 \ 0 & 22 & -6 \ 7 & -13 & 0 \end{bmatrix} \] \[ \det(A_y) = 5(22 \cdot 0 - (-6) \cdot -13) - 6(0 \cdot 0 - (-6) \cdot 7) + 1(0 \cdot -13 - 22 \cdot 7) \] \[ = 5(-78) - 6(42) - 154 \] \[ = -390 - 252 - 154 \] \[ = -796 \]**Matrix \( A_z \):**\[ A_z = \begin{bmatrix} 5 & -3 & 6 \ 0 & 4 & 22 \ 7 & 10 & -13 \end{bmatrix} \] \[ \det(A_z) = 5(4 \cdot -13 - 22 \cdot 10) - (-3)(0 \cdot -13 - 22 \cdot 7) + 6(0 \cdot 10 - 4 \cdot 7) \] \[ = 5(-52 - 220) + 3(154) - 6(28) \] \[ = -1360 + 462 - 168 \] \[ = -1066 \]
04

Apply Cramer's Rule to Find Solutions

Using Cramer's Rule, we find each variable by the formula: \[ x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}, \quad z = \frac{\det(A_z)}{\det(A)} \]Calculate each: \[ x = \frac{612}{300} = 2.04 \] \[ y = \frac{-796}{300} = -2.65 \] \[ z = \frac{-1066}{300} = -3.553 \]

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Key Concepts

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

Determinant of a Matrix
The determinant of a matrix is a special number that you can calculate from its elements. For a square matrix, like the 3x3 matrix here, the determinant provides key insights into the properties of the matrix, such as whether it is invertible or not. To calculate the determinant of a 3x3 matrix, use the formula:\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]This might look complicated at first, but it is just a series of multiplications and subtractions. Each part of this equation corresponds to different combinations of elements from the matrix. In essence, the determinant gives us a scalar value that helps in solving systems of equations using Cramer's Rule.
  • If the determinant is zero, the matrix is non-invertible, implying that the system has either no solutions or infinitely many solutions.
  • If the determinant is not zero, the system has a unique solution, which is what we are looking for in this exercise.
Knowing how to compute a determinant is a crucial skill in solving larger and more complex systems of equations.
Matrix Equation
A matrix equation is a compact way to represent a system of equations, making it easier to apply various mathematical techniques to find the solution. The general form is expressed as \(A\mathbf{x} = \mathbf{b}\), where:
  • \(A\) is the coefficient matrix, containing coefficients from the variables in the system of equations.
  • \(\mathbf{x}\) is the variable matrix, which includes variables like \(x\), \(y\), and \(z\).
  • \(\mathbf{b}\) is the constants matrix, which contains the constant terms from each equation.
Converting a system of equations into a matrix equation offers multiple benefits:
  • Streamlines the organization of equations, making it easier to visualize and strategize solutions.
  • Allows use of matrix operations to solve the equations efficiently, especially with techniques like cramer's Rule or matrix inverses.
By using matrices, we abstract the problem into a form that can be manipulated with linear algebra tools, facilitating a systematic approach to find the variables' values.
System of Equations
A system of equations consists of multiple equations that you need to solve together. In our case, the system involves three equations with three variables. It is represented as follows:- \(5x - 3y + z = 6\)- \(4y - 6z = 22\)- \(7x + 10y = -13\)The goal is to find the values of variables \(x\), \(y\), and \(z\) that satisfy all equations simultaneously. There are different methods to solve such systems:
  • Substitution and elimination: These are traditional methods taught early in algebra courses.
  • Matrix-based methods: More advanced techniques, like using the inverse of a matrix or Cramer鈥檚 Rule, require matrices and determinants.
In Cramer's Rule, for each variable, a modified version of the original matrix is created by replacing one of the columns of the coefficient matrix with the constants. You then solve each modified system to find the individual variable values. Understanding how systems of equations interact can shed light on many real-world problems, making this concept a fundamental piece of mathematics.

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Most popular questions from this chapter

Polynomials Determined by a Set of Points We all know that two points uniquely determine a line \(y=a x+b\) in the coordinate plane. Similarly, three points uniquely determine a quadratic (second-degree) polynomial $$y=a x^{2}+b x+c$$ four points uniquely determine a cubic (third-degree) polynomial $$y=a x^{3}+b x^{2}+c x+d$$ and so on. (Some exceptions to this rule are if the three points actually lie on a line, or the four points lie on a quadratic or line, and so on.) For the following set of five points, find the line that contains the first two points, the quadratic that contains the first three points, the cubic that contains the first four points, and the fourth-degree polynomial that contains all five points. $$(0,0), \quad(1,12), \quad(2,40), \quad(3,6), \quad(-1,-14)$$ Graph the points and functions in the same viewing rectangle using a graphing device.

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{l} x^{2}-y \geq 0 \\ x+y<6 \\ x-y<6 \end{array}\right. $$

Manufacturing Furniture A furniture factory makes wooden tables, chairs, and armoires. Each piece of furniture requires three operations: cutting the wood, assembling, and finishing. Each operation requires the number of hours ( \(h\) ) given in the table. The workers in the factory can provide 300 hours of cutting, 400 hours of assembling, and 590 hours of finishing each work week. How many tables, chairs, and armoires should be produced so that all available labor-hours are used? Or is this impossible? $$\begin{array}{|l|c|c|c|} \hline & \text { Table } & \text { Chair } & \text { Armoire } \\ \hline \text { Cutting (h) } & \frac{1}{2} & 1 & 1 \\ \text { Assembling (h) } & \frac{1}{2} & 1 \frac{1}{2} & 1 \\ \text { Finishing (h) } & 1 & 1 \frac{1}{2} & 2 \\ \hline \end{array}$$

Sketch the triangle with the given vertices, and use a determinant to find its area. $$(-2,5),(7,2),(3,-4)$$

Use Cramer's Rule to solve the system. $$\left\\{\begin{aligned} -2 a+c &=2 \\ a+2 b-c &=9 \\ 3 a+5 b+2 c &=22 \end{aligned}\right.$$
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