/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Use Cramer's Rule to solve each ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 31

Use Cramer's Rule to solve each system. $$\left\\{\begin{aligned}4 x-5 y-6 z &=-1 \\\x-2 y-5 z &=-12 \\\2 x-y &=7\end{aligned}\right.$$

Short Answer

Expert verified
The solution to the given system of equations using Cramer's Rule is \(x = -6.125\), \(y = -6.583\), and \(z = -2.125\)

Step by step solution

01

Write down the system of equations

The system of equations is given by:\n \(4x - 5y - 6z = -1\)\n\(x - 2y - 5z = -12\)\n\(2x - y = 7\) Note: The third equation does not have the variable \(z\), hence for this equation \(z = 0\)
02

Formulate the coefficient matrix and calculate its determinant

Form the coefficient matrix, \(A\), of the system and calculate its determinant, \(\Delta\), as follows: \[A = \begin{bmatrix} 4 & -5 & -6 \\ 1 & -2 & -5 \\ 2 & -1 & 0 \end{bmatrix}\]Then calculate the determinant of A:\[\Delta = \begin{vmatrix} 4 & -5 & -6 \\ 1 & -2 & -5 \\ 2 & -1 & 0 \end{vmatrix} = 4(-2(0)-(-1)(-5)) - (-5)(1(0)-2 * -5) - (-6)(1*-1 - 2*-2) = 4*5 - 5 * 10 - 6 * 3 = 20 - 50 - 18 = -48
03

Calculate the determinants for \(x\), \(y\) and \(z\)

Replace each column in turn by the column vector on the right of the equals sign (-1, -12, 7), and solve for the determinant in each case:For \(x\), \(\Delta_x\) replace the first column of the coefficient matrix with the vector on the right side:\[\Delta_x = \begin{vmatrix} -1 & -5 & -6 \\ -12 & -2 & -5 \\ 7 & -1 & 0 \end{vmatrix} = -1(-2*0-(-1*-5)) - (-5)(-12*0 - 7*-5) - (-6)(-12*-1 - 7*1) = -1*-5 - 5*-35 + 6*19 = 5 +175 + 114 = 294\]For \(y\), \(\Delta_y\) replace the second column of the coefficient matrix with the vector on the right side:\[\Delta_y = \begin{vmatrix} 4 & -1 & -6 \\ 1 & -12 & -5 \\ 2 & 7 & 0 \end{vmatrix} = 4*(-12*0 - 7*-5) - (-1)(1*0 - 2*-5) - (-6)(1*7 - 2*-12) = 4*35 - 1*-10 + 6*31 =140 - 10 + 186 =316\]For \(z\), \(\Delta_z\) replace the third column of the coefficient matrix with the vector on the right side:\[\Delta_z = \begin{vmatrix} 4 & -5 & -1 \\ 1 & -2 & -12 \\ 2 & -1 & 7 \end{vmatrix} = 4*(-2*7 - -1*-12) - (-5)(1*7 - 2*-12) - (-1)(1*-1 - 2*-2) = 4*-14 + 5*31 + 1*3 =-56 + 155 +3 = 102\]
04

Solve for x, y and z using Cramer's Rule

Cramer's Rule states that the solution of the system is given by \(x = \Delta_x/\Delta\), \(y = \Delta_y/\Delta\) and \(z = \Delta_z/\Delta\)Substituting for \(\Delta_x, \Delta_y, \Delta_z\) and \(\Delta\) yields:\[ x = \Delta_x/\Delta = 294/-48 = -6.125 \]\[ y = \Delta_y/\Delta = 316/-48 = -6.583 \]\[ z = \Delta_z/\Delta = 102/-48 = -2.125 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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 that share common variables. In the exercise given, we deal with three equations that involve the variables \(x\), \(y\), and \(z\). Solving such a system means finding the values of these variables that satisfy all the equations simultaneously.
The purpose of solving a system of equations is to find these unknowns, which might represent various practical situations like shadows in geometry or currents in an electrical circuit.
  • To express a system of linear equations, we often write it in a matrix form using coefficients of the variables.
  • In linear algebra, systems can be linear (straight-line relationships) or non-linear (involving curves).
In this particular problem, note that the third equation doesn't include the variable \(z\), but the system is classified as a linear system since the highest power of each variable is 1.
Determinants
The determinant is a special number that can be calculated from a square matrix. It provides important properties about the matrix and the system it represents.
In the context of systems of equations, determining whether the determinant of the coefficient matrix is zero or non-zero tells us a lot:
  • If the determinant is zero, the system may have no solutions or an infinite number of solutions.
  • If the determinant is non-zero, the system has a unique solution.
For the system of equations in our problem, the coefficient matrix has a determinant of \(\Delta = -48\), which is non-zero. This implies that the system has a unique solution for the variables \(x\), \(y\), and \(z\). Understanding how to compute the determinant is key in methods like Cramer's Rule used here.
Linear Algebra
Linear algebra is a field of mathematics focusing on vector spaces and linear mappings. It's core to solving systems of linear equations.
By representing equations as matrices and vectors, linear algebra provides powerful tools and methods:
  • Cramer's Rule, which uses determinants, to solve for variables in a linear system.
  • Matrix operations, such as addition, multiplication, and finding inverse matrices.
In practice, methods from linear algebra like those used in this problem, help in modeling real-world phenomena. From economic forecasts to engineering designs, using the principles of linear algebra provides clarity to complex systems.
Matrices
Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. They are powerful tools in representing and solving systems of linear equations.
The matrix form consists of:
  • Rows that correspond to individual equations in the system.
  • Columns corresponding to each variable鈥檚 coefficients.
In the exercise, we see a 3x3 matrix for the coefficients of \(x\), \(y\), and \(z\). By manipulating matrices through operations like finding the determinant or substituting columns, we can solve equations effectively. Matrices also highlight the power of systematic solutions, bringing order to the complex interaction of numbers. Whether it's in computers changing graphics or systems predicting weather, matrices are central to applied mathematics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{r} {2 x+y=z+1} \\ {2 x=1+3 y-z} \\ {x+y+z=4} \end{array}\right. $$

Solve the system: $$ \left\\{\begin{aligned} \ln w+\ln x+\ln y+& \ln z &=-1 \\ -\ln w+4 \ln x+\ln y-& \ln z=& 0 \\ \ln w-2 \ln x+\ln y-& 2 \ln z=& 11 \\ -\ln w-2 \ln x+\ln y+& 2 \ln z=&-3 \end{aligned}\right. $$ (Hint: Let \(A=\ln w, B=\ln x, C=\ln y,\) and \(D=\ln z .\) Solve the system for \(A, B, C,\) and \(D .\) Then use the logarithmic equations to find \(w, x, y, \text { and } z .)\)

The figure shows the letter \(L\) in a rectangular coordinate system. The figure can be represented by the matrix $$ B=\left[\begin{array}{llllll} {0} & {3} & {3} & {1} & {1} & {0} \\ {0} & {0} & {1} & {1} & {5} & {5} \end{array}\right] $$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The L is completed by connecting the last point in the matrix, (0,5), to the starting point, (0,0) . Use these ideas to solve Exercises 53-60. a. If \(A=\left[\begin{array}{rr}{1} & {0} \\ {0} & {-1}\end{array}\right],\) find \(A B\). b. Graph the object represented by matrix \(A B .\) What effect does the matrix multiplication have on the letter \(L\) represented by matrix \(B ?\)

Let $$ A=\left[\begin{array}{rr} {-3} & {-7} \\ {2} & {-9} \\ {5} & {0} \end{array}\right] \text { and } B=\left[\begin{array}{rr} {-5} & {-1} \\ {0} & {0} \\ {3} & {-4} \end{array}\right] $$ Solve each matrix equation for X. $$ 4 A+3 B=-2 X $$

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. Imagine the entire global population as a village of precisely 200 people. The bar graph shows some numeric observations based on this scenario. (graph can't copy) Combined, there are 183 Asians, Africans, Europeans, and Americans in the village. The number of Asians exceeds the number of Africans and Europeans by \(70 .\) The difference between the number of Europeans and Americans is \(15 .\) If the number of Africans is doubled, their population exceeds the number of Europeans and Americans by \(23 .\) Determine the number of Asians, Africans, Europeans, and Americans in the global village.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.