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.

Understand how to solve the ratio problem (b+c):(a+c):(a+b) = 4:5:6 using algebraic methods. This step-by-step guide involves setting up and solving systems of linear equations, using concepts like cross-multiplication and algebraic manipulation. Suitable for students in Grades 9-12.

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