We have a system of three linear equations:

1. \( a + b + c = 0 \)
2. \( 4a + 2b + c = -4 \)
3. \( 9a + 3b + c = -18 \)

### Step 1: Eliminate \( c \)

To eliminate \( c \), we will subtract the first equation from the second and the second from the third.

#### Subtract equation (1) from equation (2):
\[
(4a + 2b + c) - (a + b + c) = -4 - 0
\]
\[
3a + b = -4
\]
This is equation (4).

#### Subtract equation (2) from equation (3):
\[
(9a + 3b + c) - (4a + 2b + c) = -18 - (-4)
\]
\[
5a + b = -14
\]
This is equation (5).

### Step 2: Solve the system of equations with two variables
Now we have two equations:

1. \( 3a + b = -4 \)
2. \( 5a + b = -14 \)

Subtract equation (1) from equation (2):
\[
(5a + b) - (3a + b) = -14 - (-4)
\]
\[
2a = -10
\]
\[
a = -5
\]

### Step 3: Substitute \( a = -5 \) into one of the equations
Substitute \( a = -5 \) into equation \( 3a + b = -4 \):
\[
3(-5) + b = -4
\]
\[
-15 + b = -4
\]
\[
b = 11
\]

### Step 4: Solve for \( c \)
Now substitute \( a = -5 \) and \( b = 11 \) into the first equation \( a + b + c = 0 \):
\[
-5 + 11 + c = 0
\]
\[
6 + c = 0
\]
\[
c = -6
\]

### Final Solution:
\[
a = -5, \quad b = 11, \quad c = -6
\]

Let me know if you want any details or have questions.

Here are 5 related questions:
1. How can you verify if the solution is correct by substituting it back into the original equations?
2. What is the general method for solving systems of equations using elimination?
3. How would you solve this system using matrix methods such as Gaussian elimination?
4. Can this system be solved using substitution instead of elimination?
5. How would this problem change if the equations were non-linear?

**Tip**: To quickly check your solution, always substitute the values back into the original equations to confirm they hold true.

a + b + c = 0
4a + 2b + c = -4
9a + 3b + c = -18

Learn how to solve a system of three linear equations: a + b + c = 0, 4a + 2b + c = -4, and 9a + 3b + c = -18 using the elimination method. The solution involves eliminating variables, solving the reduced system, and back-substituting to find the values of a, b, and c. Suitable for Grades 9-12 students.

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