https://file.aiquickdraw.com/mathbot/file/1727797248653g2e0ygue.jpg

We are given a system of three linear equations:

\[
\begin{align*}
x_1 - 2x_2 + x_3 &= 0 \quad \text{(Equation 1)} \\
2x_2 - 8x_3 &= 8 \quad \text{(Equation 2)} \\
-4x_1 + 5x_2 + 9x_3 &= -9 \quad \text{(Equation 3)}
\end{align*}
\]

Let's solve this system using the substitution or elimination method. Here's the step-by-step solution:

### Step 1: Express \( x_1 \) from Equation (1)
From the first equation:
\[
x_1 = 2x_2 - x_3
\]

### Step 2: Substitute \( x_1 \) into Equation (3)
Substitute \( x_1 = 2x_2 - x_3 \) into the third equation:

\[
-4(2x_2 - x_3) + 5x_2 + 9x_3 = -9
\]

Simplify:
\[
-8x_2 + 4x_3 + 5x_2 + 9x_3 = -9
\]
\[
-3x_2 + 13x_3 = -9
\]
\[
3x_2 = 13x_3 + 9 \quad \text{(Equation 4)}
\]

### Step 3: Solve Equation (2) for \( x_2 \)
From Equation (2):
\[
2x_2 - 8x_3 = 8
\]
Solve for \( x_2 \):
\[
x_2 = 4x_3 + 4
\]

### Step 4: Substitute \( x_2 \) into Equation (4)
Now substitute \( x_2 = 4x_3 + 4 \) into Equation (4):

\[
3(4x_3 + 4) = 13x_3 + 9
\]
Simplify:
\[
12x_3 + 12 = 13x_3 + 9
\]
\[
12 - 9 = 13x_3 - 12x_3
\]
\[
x_3 = 3
\]

### Step 5: Find \( x_2 \)
Substitute \( x_3 = 3 \) into \( x_2 = 4x_3 + 4 \):

\[
x_2 = 4(3) + 4 = 12 + 4 = 16
\]

### Step 6: Find \( x_1 \)
Substitute \( x_2 = 16 \) and \( x_3 = 3 \) into \( x_1 = 2x_2 - x_3 \):

\[
x_1 = 2(16) - 3 = 32 - 3 = 29
\]

### Final Solution:
The solution to the system is:
\[
x_1 = 29, \quad x_2 = 16, \quad x_3 = 3
\]

Would you like any further details or explanations on this solution?

Here are 5 related questions to expand on this topic:
1. How do you solve systems of equations using matrix methods?
2. Can you apply the Gaussian elimination method to this system?
3. How do systems of equations relate to vector spaces and linear transformations?
4. What are the conditions for a system of linear equations to have a unique solution?
5. Can you solve this system using Cramer's rule?

**Tip**: Always check the solution by substituting the values back into the original equations to verify they hold true.

Solve the system of equations: x1 - 2x2 + x3 = 0, 2x2 - 8x3 = 8, -4x1 + 5x2 + 9x3 = -9

Solve a system of three linear equations step-by-step using the substitution method. This includes finding the values of x1, x2, and x3. Suitable for students in 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