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.