Question

If the system of equations

3x + 4y + 5z = a 4x + 5y + 6z = b 5x + 6y + 7z = c

is consistent then the value of $\frac{a+c}{2b}$ is (where a, b, c ∈ R)

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (B)

1

Answer 2

The given system of equations is:

1. $3x + 4y + 5z = a$
2. $4x + 5y + 6z = b$
3. $5x + 6y + 7z = c$

We can analyze the relationship between the equations. Consider the sum of the first and third equations: $(3x + 4y + 5z) + (5x + 6y + 7z) = a + c$ $8x + 10y + 12z = a + c$

We can factor out 2 from the left side: $2(4x + 5y + 6z) = a + c$

From the second equation, we know that $4x + 5y + 6z = b$. Substituting this into the equation above: $2(b) = a + c$ $2b = a + c$

This relationship $a+c = 2b$ is a necessary condition for the system to have a solution $(x, y, z)$.

The question states that the system is consistent, which means the condition $a+c = 2b$ holds. We are asked to find the value of $\frac{a+c}{2b}$. Since $a+c = 2b$, we can substitute this into the expression: $\frac{a+c}{2b} = \frac{2b}{2b} = 1$