https://file.aiquickdraw.com/mathbot/file/1729339248841mqppslw4.jpg

The system of linear equations in your image is as follows:

\[
\begin{aligned}
1. \quad x + y + z &= 6 \\
2. \quad x + 2y + 3z &= 14 \\
3. \quad x + 4y + 7z &= 30 \\
\end{aligned}
\]

We need to determine whether this system is consistent, meaning whether it has a solution, and then solve it if it is.

### Step 1: Write the system in matrix form
The augmented matrix corresponding to the system is:

\[
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 6 \\
1 & 2 & 3 & 14 \\
1 & 4 & 7 & 30
\end{array}\right]
\]

### Step 2: Perform row reduction
We will now use Gaussian elimination to reduce the matrix to row-echelon form.

1. Subtract the first row from the second and third rows:

\[
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 6 \\
0 & 1 & 2 & 8 \\
0 & 3 & 6 & 24
\end{array}\right]
\]

2. Subtract 3 times the second row from the third row:

\[
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 6 \\
0 & 1 & 2 & 8 \\
0 & 0 & 0 & 0
\end{array}\right]
\]

The third row is all zeros, indicating that one of the equations is dependent on the others. Therefore, the system is consistent and has infinitely many solutions.

### Step 3: Solve the system
We can express the system in terms of one of the variables, say \( z \).

From the second row:
\[
y + 2z = 8 \quad \Rightarrow \quad y = 8 - 2z
\]

From the first row:
\[
x + y + z = 6 \quad \Rightarrow \quad x + (8 - 2z) + z = 6 \quad \Rightarrow \quad x = -2 + z
\]

Thus, the general solution is:
\[
x = -2 + z, \quad y = 8 - 2z, \quad z = z
\]
where \( z \) is a free parameter.

### Conclusion:
The system is consistent and has infinitely many solutions, with the solution set given by:
\[
\boxed{x = -2 + z, \quad y = 8 - 2z, \quad z = z}
\]
where \( z \) is any real number.

Would you like further clarification or details on any part of the solution?

### Related Questions:
1. What are the conditions for a system of linear equations to be consistent?
2. How can you interpret a free parameter in a system of equations?
3. What is the difference between consistent and inconsistent systems?
4. How does Gaussian elimination help in solving systems of equations?
5. Can a system of equations have exactly one solution, and how can you tell?

### Tip:
When solving systems of linear equations, always check for dependent equations by row-reducing the augmented matrix to see if one equation can be derived from others.

Show that the system of linear equations x + y + z = 6, x + 2y + 3z = 14, x + 4y + 7z = 30 are consistent and solve them.

Learn how to determine the consistency of a system of linear equations x + y + z = 6, x + 2y + 3z = 14, and x + 4y + 7z = 30, and solve it using Gaussian elimination. This problem explores matrix operations and row reduction techniques, suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation