The given equation is a ratio involving three terms: $(b + c) : (a + c) : (a + b) = 4 : 5 : 6$. To solve for the relationship between $a$, $b$, and $c$, we can set up the following equations based on the ratios:

$\frac{b+c}{a+c} = \frac{4}{5}$ $\frac{a+c}{a+b} = \frac{5}{6}$

Step 1: Solve the first ratio

From the first ratio, $\frac{b+c}{a+c} = \frac{4}{5}$, we cross-multiply: $5(b + c) = 4(a + c)$ Expanding both sides: $5b + 5c = 4a + 4c$ Simplify by subtracting $4c$ from both sides: $5b + c = 4a$

Step 2: Solve the second ratio

From the second ratio, $\frac{a+c}{a+b} = \frac{5}{6}$, we cross-multiply: $6(a + c) = 5(a + b)$ Expanding both sides: $6a + 6c = 5a + 5b$ Simplify by subtracting $5a$ from both sides: $a + 6c = 5b$

Step 3: Solve the system of equations

Now, we have two equations:

1. $5b + c = 4a$
2. $a + 6c = 5b$

We can solve this system of equations to find the relationship between $a$, $b$, and $c$.

Do you want me to proceed with solving these equations for $a$, $b$, and $c$?

Relative Questions:

1. How do you find the relationship between three variables in a ratio problem?
2. What methods can you use to solve systems of linear equations?
3. How can you verify the solution of a ratio problem?
4. What does it mean when a ratio problem involves more than two variables?
5. How can cross-multiplication help in solving ratio problems?

Tip: Always check your solutions by substituting back into the original ratios to ensure they hold true.