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Chapter 8: Problem 43

Use Cramer's Rule to solve the system of equations. $$ \left\\{\begin{aligned} 3 x-5 y+z &=-14 \\ -3 x+7 y-4 z &=9 \\ 2 x+z &=6 \end{aligned}\right. $$

Short Answer

Expert verified
The solution to the system of equations is \( x=6 \), \( y=-3 \) and \( z=2 \).

Step by step solution

01

Formulate the System as a Matrix

First, write the system of equations in cube form. This results in the following 3x3 matrix: \[ \begin{bmatrix} 3 & -5 & 1 \ -3 & 7 & -4 \ 2 & 0 & 1 \end{bmatrix} \] for the coefficients, and the column of constants is: \[ \begin{bmatrix} -14 \ 9 \ 6 \end{bmatrix} \].
02

Compute the Determinant of the Coefficient Matrix

Calculate the determinant of the coefficient matrix. Using the formula for a 3x3 determinant, the determinant of matrix A, or \( |A| \), is: \[ (3*7*1+(-5*(-4)*2)+(1*-3*0))-((1*7*2)+(1*(-3)*(-5))+(3*0*-4))=-3 \].
03

Compute the Determinants for x, y and z

Replace each column of the matrix individually with the constants and compute the determinant. This gives us: \For x: \( |A_x| \) is \[ \begin{bmatrix} -14 & -5 & 1 \ 9 & 7 & -4 \ 6 & 0 & 1 \end{bmatrix} \] which equals -18. \For y: \( |A_y| \) is \[ \begin{bmatrix} 3 & -14 & 1 \ -3 & 9 & -4 \ 2 & 6 & 1 \end{bmatrix} \] which equals 9. \For z: \( |A_z| \) is \[ \begin{bmatrix} 3 & -5 & -14 \ -3 & 7 & 9 \ 2 & 0 & 6 \end{bmatrix} \] which equals -6.
04

Use Cramer鈥檚 rule to solve the system

Now, use Cramer's Rule, which says that each variable is its determinant divided by the determinant of the coefficient matrix. So, \( x = \frac{|A_x|}{|A|}\), \( y = \frac{|A_y|}{|A|}\) and \( z = \frac{|A_z|}{|A|}\). Filling in gives: \( x=6 \), \( y=-3 \) and \( z=2 \).

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Key Concepts

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

Determinants
Determinants play a crucial role in understanding linear algebra and systems of equations. They are a special number calculated from a matrix, particularly square matrices, and can be used to determine whether a system of equations has a unique solution.

For a 3x3 matrix used in this system of equations, the formula for the determinant involves a combination of multiplication and subtraction of the matrix's elements. It's like a mathematical recipe that gives you an important summary of the matrix possibilities.
  • The determinant helps in finding if a matrix is invertible. If it's zero, the matrix does not have an inverse, indicating no unique solution exists for the system of equations.
  • In Cramer's Rule, determinants allow us to "divide" matrices, isolating variables as solutions for systems of linear equations.
  • When solving with Cramer's Rule, compute the determinant of the coefficient matrix first, then substitute columns with constants to find determinants specific to each variable.
Understanding determinants isn't just about crunching numbers; it's about unlocking deeper insights into systems of linear equations.
3x3 Matrices
A 3x3 matrix is simply a square grid that consists of three rows and three columns. This format is especially common when dealing with systems of three linear equations. In matrices like these, each element represents a coefficient from the system of equations.

Working with 3x3 matrices is foundational for applying Cramer's Rule. Let's see why:
  • Each row of the matrix typically corresponds to one equation from the system, and each column corresponds to one variable across the equations.
  • In Cramer's Rule, you reconfigure the 3x3 matrix by switching its columns with the column of constants from the equations. This reshuffling helps to isolate variables when computing solutions.
  • The arrangement of numbers within a matrix allows for a structured approach in applying linear algebra techniques to problem-solving.
Grasping the concept of 3x3 matrices equips you with a powerful tool to translate complex systems into manageable arithmetic operations.
System of Linear Equations
A system of linear equations is essentially a collection of two or more linear equations involving the same set of variables. In this context, we're specifically dealing with three equations.

The cold-hearted, mechanical side involves algebra, but don't forget the practical aspects:
  • Each equation in the system is a straight line when graphed in a space that the variables define. The solution to the system is where these lines intersect.
  • Solving systems of linear equations means finding values for variables that satisfy all equations simultaneously. Forming a matrix is a powerful way to systematically approach these equations.
  • Cramer's Rule is an elegant strategy for solving a system when the number of equations equals the number of variables, using determinants of matrices to find solutions.
Systems of linear equations are ubiquitous in mathematics and the real world, from physics to economics, providing practical methods for modeling and solving real-life problems.

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Most popular questions from this chapter

Matrix G gives the U.S. gross domestic product for the years \(1999-2001\) GDP(billions of \(\mathfrak{S})$$\begin{array}{l}1999 \\ 2000 \\\ 2001\end{array}\left[\begin{array}{r}9274.3 \\ 9824.6 \\\ 10,082.2\end{array}\right]=G\) The finance, retail, and agricultural sectors contributed \(20 \%, 9 \%,\) and \(1.4 \%,\) respectively, to the gross domestic product in those years. These percentages have been converted to decimals and are given in matrix \(P .\) (Source: U.S. Bureau of Economic Analysis) Finance Retail Agriculture $$\left[\begin{array}{lll}0.2 & 0.09 & 0.014\end{array}\right]=P$$ (a) Compute the product \(G P\). (b) What does GP represent? (c) Is the product \(P G\) defined? If so, does it represent anything meaningful? Explain.

For the given matrices \(A, B,\) and \(C,\) evaluate the indicated expression. $$\begin{aligned}&A=\left[\begin{array}{rr}3 & -8 \\\2 & 4\end{array}\right] ; \quad B=\left[\begin{array}{rr}-6 & 0 \\\0 & -6\end{array}\right] ; \quad C=\left[\begin{array}{rr}3 & 5 \\\\-2 & 6\end{array}\right]\\\&(A+2 B) C\end{aligned}$$

Apply elementary row operations to a matrix to solve the system of equations. If there is no solution, state that the system is inconsistent. \(\left\\{\begin{array}{l}x+4 z=-3 \\ x-5 y=0 \\ z+4 y=2\end{array}\right.\) (Hint: Be careful with the order of the variables.)

Keith and two of his friends, Sam and Cody, take advantage of a sidewalk sale at a shopping mall. Their purchases are summarized in the following table. $$\begin{array}{lc|c|c|} \hline& {3}{|}\text { Quantity } \\\\\hline\text { Name } & \text { Shirt } & \text { Sweater } & \text { Jacket } \\\\\hline \text { Keith } & 3 & 2 & 1 \\\\\text { Sam } & 1 & 2 & 2 \\\\\text { Cody } & 2 & 1 & 2\\\\\hline\end{array}$$ The sale prices are \(\$ 14.95\) per shirt, \(\$ 18.95\) per sweater, and \(\$ 24.95\) per jacket. In their state, there is no sales tax on purchases of clothing. Use matrix multiplication to determine the total expenditure of each of the three shoppers.

Explain why the following system of equations has no solution. $$\left\\{\begin{aligned} (x+y)^{2} &=36 \\ x y &=18 \end{aligned}\right.$$ (Hint: Expand the expression \((x+y)^{2}\).)
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