Solving a System of Equations to Find the Value of b

The problem provides us with three equations involving variables \(a\), \(b\), and \(c\). We need to find the specific value of \(b\).

The given equations are:

• \(a(a + b + c)^2 = 1792\)
• \(b(a + b + c)^2 = 1536\)
• \(c(a + b + c)^2 = 768\)

Let's simplify the problem by introducing a new variable. Let \(S = a + b + c\). Now, we can rewrite the equations using \(S\):

• \(a \cdot S^2 = 1792\) (Equation 1)
• \(b \cdot S^2 = 1536\) (Equation 2)
• \(c \cdot S^2 = 768\) (Equation 3)

Combining the Equations

Notice that the term \(S^2\) is common in all three equations. If we add these three equations together, we can factor out \(S^2\):

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

Factor out \(S^2\) from the left side:

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

We know that \(a + b + c = S\). Substitute \(S\) back into the left side of the equation:

\(S \cdot S^2 = 1792 + 1536 + 768\)

\(S^3 = 1792 + 1536 + 768\)

Calculating the Sum and Finding S

Now, let's calculate the sum on the right side:

\(1792 + 1536 = 3328\)

\(3328 + 768 = 4096\)

So, the equation becomes:

\(S^3 = 4096\)

To find \(S\), we need to calculate the cube root of 4096:

\(S = \sqrt[3]{4096}\)

We need to find a number that, when multiplied by itself three times, equals 4096. Let's try some numbers:

• \(10^3 = 1000\)
• \(15^3 = 15 \times 15 \times 15 = 225 \times 15 = 3375\)
• \(16^3 = 16 \times 16 \times 16 = 256 \times 16 = 4096\)

So, \(S = 16\).

Since \(S = a + b + c\), we know that \(a + b + c = 16\).

Finding the Value of b

We need to find the value of \(b\). We can use any of the original equations. Let's use Equation 2, which directly involves \(b\):

\(b \cdot S^2 = 1536\)

We found that \(S = 16\). Substitute this value into the equation:

\(b \cdot (16)^2 = 1536\)

\(b \cdot 256 = 1536\)

Now, solve for \(b\) by dividing both sides by 256:

\(b = \frac{1536}{256}\)

Let's perform the division:

\(1536 \div 256\)

We can estimate. \(256 \times 5 = 1280\). \(256 \times 6 = 1536\).

So,

\(b = 6\)

The value of \(b\) is 6.

Final Answer Check

If \(b=6\) and \(S=16\), then \(b \cdot S^2 = 6 \cdot 16^2 = 6 \cdot 256 = 1536\), which matches the given equation for \(b\).

We could also find \(a\) and \(c\) if needed:

• \(a \cdot 16^2 = 1792 \implies a \cdot 256 = 1792 \implies a = \frac{1792}{256} = 7\)
• \(c \cdot 16^2 = 768 \implies c \cdot 256 = 768 \implies c = \frac{768}{256} = 3\)

Let's check if \(a + b + c = S\): \(7 + 6 + 3 = 16\), which is equal to \(S\).

The value \(b=6\) is consistent with all given equations.

Revision Table: Key Steps in Solving Equations

Understanding the steps helps in solving similar system of equations problems.

• Identify common factors or terms in the equations.
• Use substitution or combination methods to simplify.
• Solve for intermediate variables (like \(S\) in this case).
• Substitute the value of intermediate variables back into the original equations to find the desired variable.
• Check your answer using the original equations.

Additional Information: Algebraic Manipulation and Cube Roots

This problem required basic algebraic manipulation and calculating a cube root.

Algebraic Manipulation: This involves rearranging or combining equations to isolate variables. Adding equations together, factoring out common terms, and substitution are key techniques.

Cube Roots: The cube root of a number \(x\) is a number \(y\) such that \(y^3 = x\). It's denoted as \(\sqrt[3]{x}\). In our case, we needed to find \(\sqrt[3]{4096}\). Knowing common cubes (like \(10^3=1000\), \(16^3=4096\)) can be helpful.

Solving systems of equations is a fundamental concept in algebra, applicable in various fields.