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A system of equations is a collection of two or more equations with the same set of unknowns. In this lumber cutting problem, we're dealing with three unknowns: the rates at which saws A, B, and C can cut lumber. The goal is to find these rates.

The main idea is to create equations based on the conditions given in the problem. We use these equations to find the values of the unknowns. In this case, the rates of cutting lumber by each saw. We start by representing the rates with variables. For example, we let:

• A be the rate of saw A
• B be the rate of saw B
• C be the rate of saw C

Next, we set up the system of equations based on the problem statements. For instance:

• Equation 1: All saws together for 6 hours cut 7200 ft of lumber, represented as A + B + C = 1200.
• Equation 2: Saws A and B together for 9.6 hours cut 7200 ft of lumber, which we write as A + B = 750.
• Equation 3: Saws B and C together for 9 hours cut 7200 ft of lumber, or B + C = 800.

With these equations, we now have our system of equations to solve for A, B, and C.

Solving a system of equations requires algebraic methods such as substitution or elimination. Here, we use substitution primarily. First, isolate a variable in one of the equations; we chose Equation 2 (A + B = 750) and solved for A:
\[ A = 750 - B \] Then, substitute A into other equations to solve for B and C.

For instance, substitute A into Equation 1: \[ (750 - B) + B + C = 1200 \] Simplifying this equation, we derive:
\[ 750 + C = 1200 \] Solving for C:
\[ C = 450 \] Now with C isolated, use the value of C to find B in Equation 3.

The main principle here is that each variable can be isolated and substituted step by step. This simplification allows us to manage and determine the rates in a systematic and logical manner.

In this problem, rates refer to how quickly each saw can cut lumber. We often deal with rates in real-world problems, for example, speed (distance over time), growth (quantity over time), or productivity (output over time).

The key is to interpret rates correctly. For instance, the rate for saw A is found as the amount of lumber it cuts per hour (ft/hr). Similarly, B and C have their rates in the same units. We derived A = 575 ft/hr, B = 175 ft/hr, and C = 625 ft/hr.

The problem also involves ratios, as the rates are compared to each other. If you look at the equations created initially: \[ A + B + C = 1200, \] \[ A + B = 750, \] \[ B + C = 800 \] These equations express ratios of combinations of saws working together to meet the desired lumber output.

Understanding the relationships between these rates and combining them correctly is crucial in problem-solving. By mastering these fundamental concepts – systems of equations, algebra, and understanding rates – you can tackle similar problems effectively.