2(w + 3w) = 64
Solving the Equation 2(w + 3w) = 64: A Step-by-Step Guide with Common Problems Explained
Understanding and Solving the Equation 2(w + 3w) = 64
Equations are fundamental tools in algebra that help us find unknown values by balancing expressions. One common type is linear equations involving parentheses and combined like terms. An equation like 2(w + 3w) = 64 might seem simple, but mastering its solution builds a strong foundation for more complex math concepts.
If you're learning algebra or reviewing linear equations, solving something like 2(w + 3w) = 64 helps develop essential problem-solving skills. In this article, we’ll break down the steps to solve this equation, explain why each step matters, and clarify any common pitfalls students encounter.
Step 1: Simplify the Expression Inside the Parentheses
Start by combining like terms inside the parentheses: w + 3w = 4w
So the equation becomes: 2(4w) = 64
This simplification is crucial—it reduces complexity and reveals the equation’s true form: 8w = 64
Step 2: Isolate the Variable w
To solve for w, divide both sides of the equation by 2: (8w)/2 = 64/2 4w = 64
Dividing by the coefficient of w isolates the variable, making it easier to find its value. (Always divide both sides by the same number to maintain equation balance.)
Step 3: Solve for w by Dividing Both Sides by 4
Now divide both sides by 4: 4w ÷ 4 = 64 ÷ 4 w = 16
This final step gives you the value of w. Always check your solution by substituting w = 16 back into the original equation.
Step 4: Verification — Plug the Solution Back In
Substitute w = 16 into 2(w + 3w) = 64: Left side: 2(16 + 3×16) = 2(16 + 48) = 2(64) = 128 Wait — that’s not equal to 64!
But hold on—let’s double-check the arithmetic carefully:
- 3×16 = 48
- w + 3w = 16 + 48 = 64
- 2 × 64 = 128
But 128 ≠ 64. This means w = 16 does NOT satisfy the original equation.
What Went Wrong? Let’s Revisit the Steps
We correctly simplified: 2(w + 3w) = 2(4w) = 8w = 64
Then: 8w = 64 → w = 64 ÷ 8 = 8
Ah! The mistake was in division: 8w = 64 → w = 64 ÷ 8 = 8, not 16.
Correct Solution: w = 8
Here’s the corrected solution:
- Simplify: 2(w + 3w) = 2(4w) = 8w
- Set equal to 64: 8w = 64
- Divide both sides by 8: w = 64 ÷ 8 = 8
- Verify: 2(8 + 3×8) = 2(8 + 24) = 2(32) = 64 ✅ This checks out.
Why This Equation Matters
The equation 2(w + 3w) = 64 models real-life scenarios such as:
- Combining two groups of items (e.g., 2 batches where one batch holds 3 times the items of w)
- Budgeting with multiple expenses -Distance or rate problems simplified into algebra
Understanding how to isolate variables and simplify expressions prepares students for advanced algebra, physics, and engineering.
Tips for Solving Linear Equations Like This
- Step 1: Combine like terms inside parentheses to simplify the equation.
- Step 2: Use inverse operations—addition/subtraction and multiplication/division—on both sides to isolate w.
- Step 3: Always verify by substituting the solution back in.
- Step 4: Watch signs carefully—distribution followed by division requires attention to positive/negative results.
Conclusion
Solving 2(w + 3w) = 64 teaches core algebraic principles: simplification, combination of terms, and isolation of variables. The critical lesson is careful arithmetic and verification. Remember: w = 8 is the correct solution, not 16. By methodically simplifying and solving step by step—and checking your answer—you build both accuracy and confidence in algebra.
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