Understanding the Context

What Is the Equation?

Let’s start with the equation:
3x – 12 + 10 – 2x = 10

At first glance, the presence of x and constant numbers might seem overwhelming. But simple algebra helps us simplify and isolate x. By combining like terms and simplifying both sides, we transform the equation into a straightforward expression: x = 12. That’s the solution — but understanding how we reach it is what truly matters.

Step-by-Step Solution

Key Insights

Step 1: Combine Like Terms

We begin by simplifying all the numerical and variable terms on the left side. The terms involving x are 3x – 2x, and the constants are –12 + 10.

• Combine the x-terms:
  3x – 2x = x
  
• Combine the constants:
  –12 + 10 = –2

Now the equation becomes:
x – 2 = 10

Step 2: Isolate the Variable x

To solve for x, we need all x-terms on one side and constants on the other. Add 2 to both sides to move –2 away:
x – 2 + 2 = 10 + 2
x = 12

Final Thoughts

Check the Solution

Always verify by plugging x = 12 back into the original equation:
3(12) – 12 + 10 – 2(12) = 10
Calculate step by step:
→ 36 – 12 + 10 – 24 = 10
→ (36 – 12) + (10 – 24) = 10
→ 24 + (–14) = 10
→ 10 = 10 ✅

The equation checks out — our solution is correct!

Why This Equation Matters — Practical Applications

While 3x – 12 + 10 – 2x = 10 is abstract, equations like this are foundational in modeling real-world situations. They help in:

• Finance: Calculating break-even points where income equals expenses
  
• Physics: Modeling motion with constant rates
  
• Computer Science: Preparing logic for algorithms across platforms
  
• Everyday Problem-Solving: Scaling recipes, comparing billing plans, budgeting