Write the given system of equations as a matrix equation and solve using inverses. Complete parts a through c.
x 1x1+4 x 2=k1
x 2+x3=k2
3 x 1−x2−14x3=k3.  What are x 1,x2,x3 when k 1=32​,k 2−3​,and k 3=142​?

The system of equations given is:

\[
\begin{aligned}
x_1 + 4x_2 &= k_1 \\
x_2 + x_3 &= k_2 \\
3x_1 - x_2 - 14x_3 &= k_3
\end{aligned}
\]

To express this system as a matrix equation, we write it in the form:

\[
A \mathbf{x} = \mathbf{k}
\]

where:

\[
A = \begin{pmatrix}
1 & 4 & 0 \\
0 & 1 & 1 \\
3 & -1 & -14
\end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}, \quad
\mathbf{k} = \begin{pmatrix}
k_1 \\
k_2 \\
k_3
\end{pmatrix}
\]

### Part a: Write the matrix equation

The matrix equation is:

\[
\begin{pmatrix}
1 & 4 & 0 \\
0 & 1 & 1 \\
3 & -1 & -14
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
=
\begin{pmatrix}
k_1 \\
k_2 \\
k_3
\end{pmatrix}
\]

### Part b: Solve using matrix inverses

We solve for \( \mathbf{x} \) using the matrix inverse:

\[
\mathbf{x} = A^{-1} \mathbf{k}
\]

To find \( A^{-1} \), we calculate the inverse of the matrix \( A \). The inverse of a 3x3 matrix can be found using the formula for the inverse of a general matrix, or by row reduction. For simplicity, let's proceed directly with the formula.

Now, given the specific values of \( k_1 = 32 \), \( k_2 = -3 \), and \( k_3 = 142 \), we substitute these into the equation:

\[
\mathbf{k} = \begin{pmatrix}
32 \\
-3 \\
142
\end{pmatrix}
\]

After finding \( A^{-1} \), we multiply it by \( \mathbf{k} \) to obtain \( \mathbf{x} \), which consists of \( x_1 \), \( x_2 \), and \( x_3 \).

Let me calculate the inverse of \( A \) and solve for \( \mathbf{x} \).

The solution to the system of equations is:

\[
x_1 = 16, \quad x_2 = 4, \quad x_3 = -7
\]

### Part c: Conclusion
For the given values \( k_1 = 32 \), \( k_2 = -3 \), and \( k_3 = 142 \), the values of \( x_1 \), \( x_2 \), and \( x_3 \) are:

\[
x_1 = 16, \quad x_2 = 4, \quad x_3 = -7
\]

Would you like more details on the steps, or do you have any questions?

Here are 5 related questions to expand on this topic:

1. How do you find the inverse of a 3x3 matrix step by step?
2. What are the conditions for a matrix to have an inverse?
3. Can you solve this system using another method, such as Gaussian elimination?
4. How does the determinant of a matrix relate to the existence of an inverse?
5. What happens if the matrix \( A \) is singular and has no inverse?

**Tip**: When solving systems of equations using matrices, ensure the matrix is invertible by checking that its determinant is non-zero.

Write the given system of equations as a matrix equation and solve using inverses. Complete parts a through c. x1 + 4x2 = k1, x2 + x3 = k2, 3x1 - x2 - 14x3 = k3. What are x1, x2, x3 when k1 = 32, k2 = -3, and k3 = 142?

Explore how to write and solve a system of linear equations using matrix inverses. The system includes equations like x1 + 4x2 = k1, x2 + x3 = k2, and 3x1 - x2 - 14x3 = k3. Learn how to find x1, x2, and x3 when k1 = 32, k2 = -3, and k3 = 142 using matrix operations. Suitable for college-level students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation