Solve The Remaining Two Equations To Find A2

Solve the Remaining Two Equations to Find a₂: A Step-by-Step Guide

When working with systems of equations, solving for unknown variables like a₂ requires a structured approach. Imagine you’re given three equations, but two have already been solved, leaving one final equation to determine a₂. This scenario is common in fields like engineering, physics, and economics, where variables often represent real-world quantities. In this article, we’ll explore how to isolate a₂ using algebraic techniques, supported by examples and practical applications.

Understanding the Problem: What Are We Solving For?

Let’s assume you have a system of three equations with three variables: a₁, a₂, and a₃. For instance:

1. Equation 1: $ 2a₁ + 3a₂ = 10 $
2. Equation 2: $ 4a₁ - a₂ = 5 $
3. Equation 3: $ a₁ + 2a₂ + a₃ = 12 $

Suppose a₁ and a₃ have already been solved (e.g., a₁ = 2 and a₃ = 3). Your task is to solve Equation 3 for a₂. This process involves substituting known values into the equation and simplifying.

Step-by-Step Guide to Solving for a₂

Step 1: Substitute Known Values

Plug the solved values of a₁ and a₃ into Equation 3:
$ a₁ + 2a₂ + a₃ = 12 $
$ 2 + 2a₂ + 3 = 12 $

Step 2: Simplify the Equation

Combine like terms:
$ 5 + 2a₂ = 12 $

Step 3: Isolate a₂

Subtract 5 from both sides:
$ 2a₂ = 12 - 5 $
$ 2a₂ = 7 $

Step 4: Solve for a₂

Divide both sides by 2:
$ a₂ = \frac{7}{2} = 3.5 $

Scientific Explanation: Why This Works

Solving systems of equations relies on the principle of balance. Each equation represents a constraint, and the solution satisfies all constraints simultaneously. By substituting known values, we reduce the number of variables, making the problem tractable. This method, called substitution, is foundational in linear algebra and is used to model real-world systems like electrical circuits or economic models.

Example: Real-World Application

Consider a business scenario where a₁ represents fixed costs, a₂ variable costs, and a₃ profit. If fixed costs (a₁) are $2,000 and profit (a₃) is $3,000, solving for a₂ (variable costs) helps determine break-even points. Using the steps above, the business could calculate that variable costs must be $3,500 to meet revenue targets.

Common Mistakes to Avoid

1. Incorrect Substitution: Double-check values before plugging them into the equation.
2. Arithmetic Errors: Simplify step-by-step to avoid miscalculations.
3. Ignoring Units: In applied problems, ensure units (e.g., dollars, meters) are consistent.

FAQ: Frequently Asked Questions

Q: Why do we need two equations to solve for three variables?
A: Each equation provides