Which Equation Is Correctly Rewritten To Solve For X

Solving for x: How to Rewrite Equations Correctly

When you’re learning algebra, one of the first concepts that feels both powerful and confusing is solving for x. Day to day, the trick isn’t just moving terms around; it’s about applying the correct rules of equality and keeping the equation balanced. It’s the skill that turns a seemingly opaque expression into a clear answer, and mastering it opens the door to everything from basic algebra to calculus. This article walks through the fundamentals, common pitfalls, and a set of step‑by‑step examples that show exactly which equation is correctly rewritten to solve for x.

Introduction

Imagine you have an equation that looks like this:

2(3x – 5) + 4 = 10

Your goal is to isolate x on one side so you can find its value. The process involves:

1. Distributing constants across parentheses.
2. Combining like terms.
3. Moving terms from one side to the other.
4. Dividing or multiplying by a coefficient to solve for x.

If you apply these steps correctly, you’ll rewrite the equation in a form that clearly shows x on one side and a number on the other: x = 2. That's why the key question is: *Which of these rewritten forms is correct? * Let’s explore the rules that guarantee a correct rewrite.

The Rules of a Correct Rewrite

1. Preserve Equality

Every operation you apply to one side of the equation must be mirrored on the other side. This keeps the two sides equal. For example:

• Adding 3 to the left side must be accompanied by adding 3 to the right side.
• Multiplying the left side by 5 must be mirrored by multiplying the right side by 5.

2. Use Inverse Operations

To isolate x, you often need to undo an operation that’s currently surrounding x. Inverse operations are:

• Addition ↔ Subtraction
• Multiplication ↔ Division
• Exponentiation ↔ Root extraction

Doing the inverse on both sides keeps the balance intact.

3. Keep the Equation Simplified

After each step, simplify the expression on both sides. Combine like terms, reduce fractions, and eliminate parentheses. A simplified equation is easier to read and less prone to error.

4. Avoid Changing the Sign of an Entire Term

When you move a term from one side to the other, remember to change its sign. Take this case: moving (+5) from the left to the right becomes (-5) on the right.

Common Mistakes and How to Spot Them

Step‑by‑Step Example 1

Equation:
[ 4x - 7 = 2(3x + 1) ]

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Goal: Solve for x.

Correct rewritten form:
[ x = -\frac{9}{2} ]

Step‑by‑Step Example 2

Equation:
[ \frac{3x + 5}{4} = 7 ]

Goal: Solve for x.

Correct rewritten form:
[ x = \frac{23}{3} ]

Step‑by‑Step Example 3

Equation with parentheses and fractions:
[ \frac{2x}{3} + \frac{1}{6} = \frac{x + 4}{2} ]

Goal: Solve for x.

Correct rewritten form:
[ x = 11 ]

Frequently Asked Questions (FAQ)

1. What if the equation has a variable on both sides?

Answer: Combine like terms on each side first, then move all variable terms to one side and constants to the other. The process is the same; you’re just doing more bookkeeping.

2. How do I handle equations with radicals or exponents?

Answer: Apply the inverse operation that will eliminate the radical or exponent. For a square root, square both sides. For an exponent, take the appropriate root or use logarithms for non‑integer exponents.

3. I keep getting a negative value for x. Is that a mistake?

Answer: Not necessarily. If the algebraic steps are correct, a negative solution is valid. Double‑check by substituting the value back into the original equation.

4. Can I add or subtract a term from only one side?

Answer: No. Every addition or subtraction must be mirrored on the opposite side to keep the equality true.

5. Why does it matter if I simplify before or after moving terms?

Answer: Simplifying early can reduce errors and make the algebra cleaner. Even so, the final result will be the same if you follow the rules correctly, regardless of order.

Conclusion

The heart of solving for x lies in maintaining equality while applying the correct inverse operations. By distributing, combining like terms, moving terms with sign changes, and finally dividing or multiplying, you can rewrite any linear equation into a form that clearly shows x alone on one side. Remember to mirror every operation across the equals sign, keep track of signs, and simplify along the way.

When you’re uncertain whether a rewrite is correct, test it: substitute the solved value of x back into the original equation. If both sides balance, your rewrite was accurate. Mastering these steps turns algebra from a series of confusing manipulations into a logical, step‑by‑step process that yields clear, reliable answers. Happy solving!