The image contains a set of mathematical equations. Let me extract and analyze the equations to solve the problem.

Extracted Equations

1. $3a + 7b + c = 315$
2. $4a + 10b + c = 420$
3. Question: Find $a + b + c$.

Let's solve this step by step.

Step 1: Eliminate $c$

Subtract the first equation from the second: $(4a + 10b + c) - (3a + 7b + c) = 420 - 315$ a + 3b = 105 \tag{3}

Step 2: Solve for $a + b + c$

From Equation (3), we know $a + 3b = 105$. We will express $c$ in terms of $a$ and $b$ using one of the original equations. From the first equation: c = 315 - 3a - 7b \tag{4}

Now substitute $a + b + c = ?$ using the above expressions: $a + b + c = a + b + (315 - 3a - 7b)$ $a + b + c = 315 - 2a - 6b$

Substitute $a = 105 - 3b$ from Equation (3) into this expression: $a + b + c = 315 - 2(105 - 3b) - 6b$ $a + b + c = 315 - 210 + 6b - 6b$ $a + b + c = 105$

Final Answer:

$a + b + c = 105$

Would you like further clarification or details about the steps?

Related Questions

1. How can we solve such equations using matrices?
2. What is the significance of eliminating variables in systems of equations?
3. How would the solution change if additional equations were added?
4. How can we check the consistency of the solutions?
5. Can this system be solved graphically?

Tip:

For solving systems of equations, always simplify and reduce the number of variables step by step for clarity and accuracy.