/*! 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 13 Using Cramer's Rule In Exercises... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 13

Using Cramer's Rule In Exercises \(7-16,\) use Cramer's Rule to solve (if possible) the system of equations. $$\left\\{\begin{array}{rr}{4 x-y+z=} & {-5} \\ {2 x+2 y+3 z=} & {10} \\ {5 x-2 y+6 z=} & {1}\end{array}\right. $$

Short Answer

Expert verified
The solutions are \( x = -53/29, y = 165/29, z = 118/29 \)

Step by step solution

01

Finding the Determinant of the Coefficient Matrix

Make the coefficient matrix A by taking only the coefficients of x, y and z from the system and evaluate its determinant. Matrix A is:\[ A = \begin{bmatrix}4 & -1 & 1 \2 & 2 & 3 \5 & -2 & 6\end{bmatrix} \]The determinant of A, denoted as |A| is\( |A| = 4(2*6 - 3*(-2)) - (-1)(2*6 - 5*3) + 1*(2*(-2) - 2*5) = 56 + 3 - 30 = 29 \)
02

Finding the Determinants of the 'x', 'y' and 'z' Matrices

Make matrix Ax by replacing the x coefficients in A with the constants on the right of the equations, Ay by replacing the y coefficients and Az by replacing the z coefficients, then find their determinants.\[ Ax = \begin{bmatrix}-5 & -1 & 1 \10 & 2 & 3 \1 & -2 & 6\end{bmatrix} \]\[ |Ax| = -5(2*6 - 3*(-2)) - (-1)(10*6 - 1*3) + 1*(10*(-2) - 2*1) = -90 + 57 - 20 = -53 \]\[ Ay = \begin{bmatrix}4 & -5 & 1 \2 & 10 & 3 \5 & 1 & 6\end{bmatrix} \]\[ |Ay| = 4(10*6 - 3*1) - (-5)(2*6 - 5*3) + 1*(2*1 - 10*5) = 228 - 15 - 48 = 165 \]\[ Az = \begin{bmatrix}4 & -1 & -5 \2 & 2 & 10 \5 & -2 & 1\end{bmatrix} \]\[ |Az| = 4(2*1 - 10*(-2)) - (-1)(2*1 - 5*10) + (-5)*(2*(-2) - 2*5) = 100 + 48 - 30 = 118 \]
03

Applying Cramer's Rule to Find 'x', 'y' and 'z'

According to Cramer's Rule,\[ x = |Ax| / |A| = -53 / 29 \]\[ y = |Ay| / |A| = 165 / 29 \]\[ z = |Az| / |A| = 118 / 29 \]

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.

Systems of Linear Equations
When we speak about systems of linear equations, we are referring to a collection of two or more linear equations with the same set of variables. The goal is to find the values of these variables that satisfy all equations in the system simultaneously.

For instance, in our exercise, we have a system of three equations with three unknowns (x, y, and z). Solving such a system can be done in various ways, including graphically, using substitution or elimination, and, as highlighted by the given problem, by using Cramer's Rule, which is an algebraic method relying on the determinant of matrices.
Determinant of a Matrix
The determinant of a matrix is a special number that is calculated from its elements. In case of a 2x2 matrix, the determinant is found by subtracting the product of its diagonals. However, as the size of the matrix increases, the calculation becomes more complex.

For a 3x3 matrix, as seen in our example, the determinant can be calculated using the 'Rule of Sarrus' or a method involving cross-multiplication and subtraction involving the expansion of minors and cofactors. For the coefficient matrix A in our exercise, the determinant is found to be 29.
Coefficient Matrix
A coefficient matrix is constructed from a system of linear equations by taking only the coefficients of the variables. This matrix is a key element in matrix algebra and is used to represent the system in a compact form.

In the system from our exercise, the coefficient matrix A is a 3x3 matrix containing the coefficients of x, y, and z in each equation. Matrix algebra allows us to manipulate this matrix to find solutions for the variables in the system, as is done when applying Cramer's Rule.
Matrix Algebra
Matrix algebra encompasses operations that can be performed on matrices, such as addition, subtraction, multiplication, and finding the inverse or the determinant. It鈥檚 an essential part of linear algebra, and it's used extensively to solve systems of linear equations.

In Cramer's Rule, matrix algebra is employed to calculate the determinants of the coefficient matrix and the modified matrices, which have one column replaced by the constants from the right-hand side of the equations. The determinants of these matrices are then used to find the values of x, y, and z in a manner that is typically easier than other methods for larger systems.

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

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. $$\left\\{\begin{aligned} x+2 y &=0 \\ x+y &=6 \\ 3 x-2 y &=8 \end{aligned}\right.$$

Interpreting Reduced Row-Echelon Form , an augmented matrix that represents a system of linear equations (in variables \(x, y,\) and \(z,\) if applicable) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$\left[ \begin{array}{rrrrr}{1} & {0} & {0} & {\vdots} & {5} \\ {0} & {1} & {0} & {\vdots} & {-3} \\ {0} & {0} & {1} & {\vdots} & {0}\end{array}\right]$$

In Exercises \(89-94\) evaluate the determinant in which the entries are functions. Determinants of this type occur when change of variables are made in calculus. $$\left| \begin{array}{ll}{x} & {\ln x} \\ {1} & {1 / x}\end{array}\right|$$

A corporation has three factories, each of which manufactures acoustic guitars and electric guitars. The production levels are represented by \(A\) . \(A=\left[ \begin{array}{ccc}{70} & {50} & {25} \\ {35} & {100} & {70}\end{array}\right]\) (a) Interpret the value of \(a_{22}\) (b) How could you find the production levels when production is increased by 20\(\% ?\) (c) Each acoustic guitar sells for \(\$ 80\) and each electric guitar sells for \(\$ 120 .\) How could you use matrices to find the total sales value of the guitars produced at each factory?

Breeding Facility A city zoo borrowed \(\$ 2,000,000\) at simple annual interest to construct a breeding facility. Some of the money was borrowed at \(8 \%,\) some at \(9 \%,\) and some at 12\(\% .\) Use a system of linear equations to determine how much was borrowed at each rate given that the total annual interest was \(\$ 186,000\) and the amount borrowed at 8\(\%\) was twice the amount borrowed at 12\(\% .\) Solve the system of linear equations using matrices.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

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.