Problem 67 - Intermediate Algebra

Question

Use matrices to solve the system. $$ \left\\{\begin{array}{l} x^{2}+y^{2}+z^{2}=14 \\ 2 x^{2}+3 y^{2}-2 z^{2}=-7 \\ x^{2}-5 y^{2}+z^{2}=8 \end{array}\right. $$

Step-by-Step Solution

Answer

Solve using matrix inversion and multiplication to find square values, then take the square root to solve for \(x\), \(y\), and \(z\).

Step 1: Represent Equations as a Matrix Equation

Rearrange the given system of equations to isolate each variable term and express it as a matrix equation. The system is:\[ x^2 + y^2 + z^2 = 14 \]\[ 2x^2 + 3y^2 - 2z^2 = -7 \]\[ x^2 - 5y^2 + z^2 = 8 \]These can be represented with matrices as follows:\[\begin{bmatrix}1 & 1 & 1 \2 & 3 & -2 \1 & -5 & 1 \end{bmatrix}\begin{bmatrix}x^2 \y^2 \z^2\end{bmatrix} = \begin{bmatrix}14 \-7 \8\end{bmatrix}\]

Step 2: Find the Inverse of the Coefficient Matrix

To solve for the variable matrix \( \begin{bmatrix} x^2 \ y^2 \ z^2 \end{bmatrix} \), we need the inverse of the coefficient matrix \( A \), where:\[A = \begin{bmatrix}1 & 1 & 1 \2 & 3 & -2 \1 & -5 & 1\end{bmatrix}\]Use a calculator or software to compute the inverse of \( A \), such that \( A^{-1} \).Ensure that the determinant of \( A \) is non-zero, otherwise the matrix is not invertible.

Step 3: Perform Matrix Multiplication to Solve for Variables

Multiply the inverse of the coefficient matrix \( A^{-1} \) by the constant matrix \( \begin{bmatrix} 14 \ -7 \ 8 \end{bmatrix} \) to find \( \begin{bmatrix} x^2 \ y^2 \ z^2 \end{bmatrix} \):\[\begin{bmatrix}x^2 \y^2 \z^2\end{bmatrix} = A^{-1} \begin{bmatrix}14 \-7 \8\end{bmatrix}\]Calculate each element of the resulting matrix to find the values for \( x^2 \), \( y^2 \), and \( z^2 \).

Step 4: Solve for x, y, z

Once you have \( x^2 \), \( y^2 \), and \( z^2 \) from the previous step, take the square roots to solve for \( x \), \( y \), and \( z \). Remember that each square has two possible roots: a positive and a negative value. Consider both possibilities when solving, but typically, one set of solutions will make sense in the context of a problem.

Key Concepts

System of Equations

A **system of equations** refers to a set of equations with multiple variables that you need to solve simultaneously. In our scenario, the variables are squared, forming what's specifically called a "nonlinear system." However, we approach nonlinear systems quite like the linear ones, by transforming them into a manageable form.
Representing these equations in matrix form helps streamline the solution process even for nonlinear scenarios. The key advantages include:

• Simplicity in managing multiple equations and variables.
• Clear visualization of coefficients and constants.
• A unified approach to solving using well-known algebraic tools.

For the given problem:- We have three equations: - Equation 1: \( x^2 + y^2 + z^2 = 14 \) - Equation 2: \( 2x^2 + 3y^2 - 2z^2 = -7 \) - Equation 3: \( x^2 - 5y^2 + z^2 = 8 \)These equations were aligned into a matrix format, setting the stage for matrix operations needed for a solution.

Matrix Inversion

Matrix inversion is a key step when solving systems of equations with matrices. This process helps us essentially "unwrap" the system and find the values of our variables. Think of the inverse of a matrix as a sort of reciprocal that, when multiplied by the original, results in the identity matrix.
To solve our nonlinear system, the matrix equation from the system of equations is:\[A\begin{bmatrix} x^2 \ y^2 \ z^2 \end{bmatrix} = \begin{bmatrix} 14 \ -7 \ 8 \end{bmatrix}\]
We found the matrix \( A \) to be a 3x3 matrix containing the coefficients:\(A = \begin{bmatrix} 1 & 1 & 1 \ 2 & 3 & -2 \ 1 & -5 & 1 \end{bmatrix}\)
Before inverting, it's vital to confirm \( A \)'s determinant is not zero, ensuring it's invertible. The inverse, \( A^{-1} \), can then be computed using a calculator or algebraic software.

• By multiplying \( A^{-1} \) with the constant matrix, we solve for the squared variables in one efficient step.
• This powerful technique provides clarity and reduces error susceptibility.

Square Roots

After solving the matrix equation using the inverse and gaining values for \( x^2 \), \( y^2 \), and \( z^2 \), the next step is finding the actual variable values through **square roots**. This is where the nonlinear aspect of our system is resolved.
Square rooting involves:

• Taking the square root of each squared variable to find \( x \), \( y \), \( z \).
• Considering both positive and negative roots, simplifying back to the context of our original problem.

For instance, if \( x^2 = 9 \), then \( x = 3 \) or \( x = -3 \).
While both values are mathematically valid, practical context or additional constraints may favor one. Selecting the appropriate root solidifies the solution, culminating in an effective comprehension of the system solved. Remembering this can prevent potential mistakes in scenarios where only integer or positive numbers fit the real-world context.