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Chapter 9: Problem 93

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} &4 x-3 y+z+1=0\\\ &\begin{array}{l} 5 x+7 y+2 z+2=0 \\ 3 x-5 y-z-1=0 \end{array} \end{aligned}$$

Short Answer

Expert verified
-67

Step by step solution

01

- Write the system of equations in standard form

Rewrite the given system of equations as: 1 = 4x - 3y + z 2 = 5x + 7y + 2z 3 = 3x - 5y - z
02

- Set up the coefficient matrix

Extract the coefficients of x, y, and z from the standard form and set up the coefficient matrix: A = [ 4 -3 1 5 7 2 3 -5 -1 ]
03

- Find the determinant of the coefficient matrix

Compute the determinant of matrix A. In this case: D = det(A) = 4(7(-1) - 2(-5)) - (-3)(5(-1) - 2(3)) + 1(5(-5) - 7(3)) D = 4(-7+10) - (-3)(-5-6) + 1(-25-21) = 4*3 - (-3)*(-11) + 1*(-46) = 12 - 33 - 46 D = - 67

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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 a common set of variables. For example, in our exercise, we have the following system:

$$\begin{aligned} 4x - 3y + z + 1 = 0 \ 5x + 7y + 2z + 2 = 0 \ 3x - 5y - z - 1 = 0 \ \ \ \text{In the above equations, x, y, and z are the variables.} \text{The goal is to find values of x, y, and z that satisfy all the equations simultaneously.} \text {There can be three types of solutions:}
  • Unique solution: Exactly one set of values for the variables.

  • No solution: The equations are inconsistent and no single set of values satisfies all the equations.

  • Infinite solutions: There are countless sets of values that satisfy all the equations.

To solve these types of problems, we often employ techniques like graphing, substitution, elimination, or matrix methods like Cramer's rule.
Determinant
The determinant is a special number that you can calculate from a square matrix. It's a useful tool in linear algebra and helps in solving systems of equations.

\text{For a 3x3 matrix like the coefficient matrix in our example:} $$\begin{bmatrix} 4 & -3 & 1 \ 5 & 7 & 2 \ 3 & -5 & -1 \ \ \ \text (The determinant, D, can be calculated as follows:)} D = det(A) = 4(7(-1) - 2(-5)) - (-3)(5(-1) - 2(3)) + 1(5(-5) - 7(3)) D = 4(-7+10) - (-3)(-5-6) + 1(-25-21) = 4*3 - (-3)*(-11) + 1*(-46) = 12 - 33 - 46 \text (Understanding and calculating the determinant is crucial since it gives us information about the system's solutions. If the determinant, \(D=0\), it means there is no unique solution to the system.)
Coefficient Matrix
The coefficient matrix contains only the coefficients of the variables in a system of linear equations. From our given system:

$$\begin{aligned} 4x - 3y + z + 1 = 0 \ 5x + 7y+ 2z + 2 = 0 \ 3x - 5y - z - 1 = 0 \ \ \ (Extracting just the coefficients of x, y, and z, we get the following matrix:)$$ \begin{bmatrix} 4 & -3 & 1 \ 5 & 7 & 2 \ 3 & -5 & -1\ \ (We use this coefficient matrix to apply Cramer's rule and find the determinant.) \

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Most popular questions from this chapter

Use a system of equations to solve each problem. Find the equation of the line \(y=a x+b\) that passes through the points \((3,-4)\) and \((-1,4)\)

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{array}{c} 12 x+8 y=3 \\ 1.5 x+y=0.9 \end{array}$$

Concept Check Find \(A B\) and \(B A\) for $$ A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$ What do you notice? Matrix \(B\) acts as the multiplicative ___________ element for \(2 \times 2\) square matrices.

Solve each linear programming problem. Aid to Disaster Victims An agency wants to ship food and clothing to tsunami victims in Japan. Commercial carriers have volunteered to transport the packages, provided they fit in the available cargo space. Each 20 - ft \(^{3}\) box of food weighs 40 lb and each \(30-\mathrm{ft}^{3}\) box of clothing weighs 10 lb. The total weight cannot exceed \(16,000 \mathrm{Ib},\) and the total volume must be at most \(18,000 \mathrm{ft}^{3} .\) Each carton of food will feed 10 people, and each carton of clothing will help 8 people. (a) How many cartons of food and clothing should be sent to maximize the number of people assisted? (b) What is the maximum number assisted? PICTURE CANT COPY

Solve each problem. Tire Sales The number of automobile tire sales is dependent on several variables. In one study the relationship among annual tire sales \(S\) (in thousands of dollars), automobile registrations \(R\) (in millions), and personal disposable income \(I\) (in millions of dollars) was investigated. The results for three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b R+c l $$ where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Jarrett, J., Business Forecasting Methods, Basil Blackwell, Ltd.) (a) Substitute the values for \(S, R,\) and \(I\) for each year from the table into the equation \(S=a+b R+c I,\) and obtain three linear equations involving \(a, b,\) and \(c\) (b) Use a graphing calculator to solve this linear system for \(a, b,\) and \(c .\) Use matrix inverse methods. (c) Write the equation for \(S\) using these values for the coefficients. (d) If \(R=117.6\) and \(I=310.73,\) predict \(S .\) (The actual value for \(S\) was \(11,314 .\) ) (e) If \(R=143.8 \text { and } I=829.06, \text { predict } S . \text { (The actual value for } S \text { was } 18,481 .)\) $$\begin{array}{|c|c|c|} \hline S & R & I \\ \hline 10,170 & 112.9 & 307.5 \\\ \hline 15,305 & 132.9 & 621.63 \\ \hline 21,289 & 155.2 & 1937.13 \\\ \hline \end{array}$$
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