Understanding the Problem: Solving for 'a'

The question provides us with three equations involving three variables, a, b, and c. Each equation has the term (a + b + c) squared. We are given the results of multiplying a, b, and c respectively by this squared term. Our goal is to find the specific numerical value of the variable 'a'.

Step-by-Step Solution

Let's write down the given equations clearly:

1. $\text{a(a + b + c)}^2 = 1792$
2. $\text{b(a + b + c)}^2 = 1536$
3. $\text{c(a + b + c)}^2 = 768$

Notice that the term (a + b + c)² is common in all three equations. This suggests we can simplify the problem by treating (a + b + c) as a single quantity.

Simplifying the Equations

Let S = a + b + c. Substituting S into the equations, we get:

1. $\text{a} \cdot \text{S}^2 = 1792$
2. $\text{b} \cdot \text{S}^2 = 1536$
3. $\text{c} \cdot \text{S}^2 = 768$

Combining the Equations

If we add all three of these simplified equations together, we can see a pattern:

$(a \cdot S^2) + (b \cdot S^2) + (c \cdot S^2) = 1792 + 1536 + 768$

We can factor out the common term S² from the left side:

$S^2 \cdot (a + b + c) = 1792 + 1536 + 768$

Remember that we defined S = a + b + c. So, the left side becomes:

$S^2 \cdot S = S^3$

Now, let's add the numbers on the right side:

$1792 + 1536 + 768 = 4096$

So, the combined equation is:

$S^3 = 4096$

Solving for S

To find the value of S, we need to calculate the cube root of 4096.

$S = \sqrt[3]{4096}$

We can test some numbers. Let's try 10³ = 1000, 20³ = 8000. The value should be between 10 and 20. Let's try 16.

$16 \times 16 \times 16 = 256 \times 16 = 4096$

So, S = 16.

Finding the Value of 'a'

Now that we know S = 16, we can use the first original equation or its simplified form:

$\text{a} \cdot \text{S}^2 = 1792$

Substitute S = 16 into this equation:

$\text{a} \cdot (16)^2 = 1792$

$\text{a} \cdot 256 = 1792$

Now, solve for 'a' by dividing 1792 by 256:

$a = \frac{1792}{256}$

To perform this division, we can think about how many times 256 fits into 1792. We know $256 \times 10 = 2560$. Let's try multiplying 256 by smaller numbers. We can estimate $1792 / 250 \approx 1800 / 250 = 180/25 = 36/5 = 7.2$. So the answer should be close to 7.

Let's calculate $256 \times 7$:

$256 \times 7 = (250 + 6) \times 7 = 250 \times 7 + 6 \times 7 = 1750 + 42 = 1792$

So, $1792 / 256 = 7$.

Thus, the value of a is 7.

Verification (Optional)

We found S = 16 and a = 7. We can find b and c similarly:

• $\text{b} \cdot \text{S}^2 = 1536 \implies \text{b} \cdot 256 = 1536 \implies b = \frac{1536}{256} = 6$
• $\text{c} \cdot \text{S}^2 = 768 \implies \text{c} \cdot 256 = 768 \implies c = \frac{768}{256} = 3$

Now check if a + b + c = S:

$7 + 6 + 3 = 16$

This matches our value for S, confirming our calculations are correct.

Final Answer

The value of 'a' is 7.

Revision Table: Key Steps in Solving Equations

Additional Information: Solving Algebraic Equations

Solving algebraic equations involves finding the values of variables that make the equations true. When you have a system of equations (more than one equation with multiple variables), you often need to use techniques like substitution or elimination to solve for the variables.

In this problem, we used a form of the elimination method by adding the equations together. This allowed us to group the terms with S² and simplify the problem into finding S first.

Understanding powers and roots (like squaring, cubing, square roots, cube roots) is also essential for solving many algebraic problems like this one. A cube root is the number that, when multiplied by itself three times, gives the original number.