3a - 5 = 8 - 2a
Solving 3a – 5 = 8 – 2a: A Step-by-Step Guide
Solving linear equations can seem daunting at first, but with a structured approach, it becomes straightforward. One common problem students encounter is solving equations like 3a – 5 = 8 – 2a. Whether you're a high school student, a homeschooler, or just learning algebra, mastering this type of equation is essential. In this article, we’ll walk through how to solve 3a – 5 = 8 – 2a step by step, explain key algebraic concepts, and provide tips to build confidence in solving similar problems.
Understanding the Equation: What Does 3a – 5 = 8 – 2a Mean?
The equation 3a – 5 = 8 – 2a contains an unknown variable a on both sides—and different coefficients (3 and –2) and constants (5 and 8). This type of equation is called a linear equation in one variable, where the variable appears only once to the first power.
Our goal is to isolate a to determine its value. This requires applying algebraic operations to both sides while maintaining equality.
Step-by-Step Solution to 3a – 5 = 8 – 2a
Step 1: Eliminate parentheses and combine like terms
The equation has no parentheses, but we can isolate a by gathering all terms containing the variable on one side and constants on the other.
Start with: 3a – 5 = 8 – 2a
Add 2a to both sides to bring all a terms together: 3a + 2a – 5 = 8 5a – 5 = 8
Now the variable a is on the left, and constants/day coefficients are on the right.
Step 2: Isolate the constant terms
Subtract 5 from both sides to eliminate the constant on the left: 5a – 5 – 5 = 8 – 5 5a = 3
Step 3: Solve for a
Now divide both sides by 5 to isolate a: a = 3 ÷ 5 a = 0.6
Final Answer
✅ The solution to the equation 3a – 5 = 8 – 2a is a = 0.6 (or a = 3/5 in fraction form).
Key Takeaways from Solving 3a – 5 = 8 – 2a
1. Balance is Key
Every operation you perform—adding, subtracting, multiplying—must be applied to both sides. This preserves the equation’s equality.
2. Order of Operations Matters
Shift variables to one side and constants to the other. Add or subtract before multiplying or dividing.
3. Verify the Solution
Plug a = 0.6 back into the original equation to confirm: Left: 3(0.6) – 5 = 1.8 – 5 = –3.2 Right: 8 – 2(0.6) = 8 – 1.2 = 6.8 – no—wait! That’s incorrect. Let’s recalculate:
Left: 3(0.6) – 5 = 1.8 – 5 = –3.2 Right: 8 – 2(0.6) = 8 – 1.2 = 6.8
Wait—this doesn’t match! Did we make a mistake?
Wait—No! That suggests a calculation error. Let’s double-check:
We had: 3a – 5 = 8 – 2a After combining: 5a – 5 = 8 Then 5a = 13? No—5a = 8 + 5 = 13? Let's re-simplify carefully:
Original: 3a – 5 = 8 – 2a Add 2a to both sides: 5a – 5 = 8 Add 5: 5a = 13 Then: a = 13/5 = 2.6
✅ So earlier steps had arithmetic errors. Let’s correct and verify fully.
Correct Full Solution (Without Errors)
3a – 5 = 8 – 2a
-
Add 2a to both sides: 3a + 2a – 5 = 8 5a – 5 = 8
-
Add 5 to both sides: 5a = 13
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Divide by 5: a = 13/5 = 2.6 or √
Check:
Left: 3(2.6) – 5 = 7.8 – 5 = 2.8 Right: 8 – 2(2.6) = 8 – 5.2 = 2.8 ✅
✔ Correct!
Why Learning to Solve This Matters
Understanding how to solve 3a – 5 = 8 – 2a gives you foundational skills for:
- Mastering more complex algebra
- Preparing for college-level math
- Solving real-world problems (e.g., budgeting, physics, finance)
Tips to Master Linear Equations
- Always keep it balanced—what you do to one side, do to the other.
- Practice rearranging terms: move constants to one side, variables to the other.
- Step through each operation logically, one at a time.
- Always verify your answer by substituting back.
- Use a sheet to track moves if stuck—pattern recognition helps.
Conclusion
Solving equations like 3a – 5 = 8 – 2a requires logical steps, attention to detail, and consistent practice. By isolating variables and eliminating terms systematically, you unlock the value of a confidently. This core skill powers success across mathematics and STEM fields.
Ready to try more? Gather your pencil, grab a notepad, and solve 2x – 7 = 5 – 3x — the process stays the same!
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