https://file.aiquickdraw.com/mathbot/file/1725370080402836f966j.jpg

The image contains a system of linear equations in three variables \(a\), \(b\), and \(c\). Let's analyze the equations provided:

1. \(a + 2b + c = 1\)
2. \(a + b + c = 2\)
3. \(2b + 3c = 4\)

We need to determine the values of \(a\), \(b\), and \(c\) that satisfy all three equations. Let's solve the system step-by-step.

### Step 1: Solve for \(a\)

First, let's subtract the second equation from the first:

\[
(a + 2b + c) - (a + b + c) = 1 - 2
\]

Simplifying:

\[
a - a + 2b - b + c - c = -1
\]

\[
b = -1
\]

### Step 2: Substitute \(b = -1\) into the equations

Substitute \(b = -1\) into the second equation:

\[
a + (-1) + c = 2
\]

\[
a - 1 + c = 2
\]

\[
a + c = 3 \quad \text{(Equation 4)}
\]

Substitute \(b = -1\) into the third equation:

\[
2(-1) + 3c = 4
\]

\[
-2 + 3c = 4
\]

\[
3c = 6
\]

\[
c = 2
\]

### Step 3: Substitute \(c = 2\) into Equation 4

Substitute \(c = 2\) into Equation 4:

\[
a + 2 = 3
\]

\[
a = 1
\]

### Solution

The values of \(a\), \(b\), and \(c\) are:

\[
a = 1, \quad b = -1, \quad c = 2
\]

Would you like more details or have any questions?

### Related Questions:

1. How can you check if a solution to a system of linear equations is correct?
2. What methods can be used to solve a system of linear equations?
3. How does the substitution method work in solving a system of equations?
4. What are the applications of solving linear equations in real-life scenarios?
5. How can systems of equations be represented graphically?

**Tip:** When solving systems of linear equations, always ensure to substitute back into the original equations to verify the solution is correct.

Tentukan nilai a, b, dan c dari sistem persamaan berikut: a + 2b + c = 1, a + b + c = 2, 2b + 3c = 4.

Determine the values of a, b, and c by solving the system of linear equations: a + 2b + c = 1, a + b + c = 2, and 2b + 3c = 4. This problem involves key algebraic concepts and is suitable for students in Grades 9-11.

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