Question: Solve for $ x $ in the equation $ 4(x - 3) + 2 = 2(2x + 1) - 6 $

February 22, 2026

Solving Linear Equations: Step-by-Step Guide to Solve for $ x $

Finding the value of $ x $ in a linear equation is a fundamental skill in algebra that helps build problem-solving confidence. One common equation students encounter is:

$$ 4(x - 3) + 2 = 2(2x + 1) - 6 $$

In this article, we’ll walk through how to solve this equation step by step, making it clear how simplifying both sides and isolating $ x $ leads to the correct solution.

Step 1: Expand Both Sides

Start by applying the distributive property to both sides of the equation:

$$ 4(x - 3) + 2 = 2(2x + 1) - 6 $$ $$ 4x - 12 + 2 = 4x + 2 - 6 $$

Now simplify both sides:

Left side: $$ 4x - 10 $$ Right side: $$ 4x - 4 $$

So the equation becomes: $$ 4x - 10 = 4x - 4 $$

Step 2: Eliminate Common Terms

Notice that $ 4x $ appears on both sides. Subtract $ 4x $ from both sides to eliminate the variable term temporarily:

$$ 4x - 10 - 4x = 4x - 4 - 4x $$ $$ -10 = -4 $$

Step 3: Analyze the Result

We now have the false statement: $$ -10 = -4 $$

This is not true, meaning there is no solution to the original equation.

What Does This Mean?

When simplifying both sides leads to a contradiction like $-10 = -4$, it indicates that the equation has no solution—it’s inconsistent. In other words, no value of $ x $ satisfies the original equation.

Recap of Steps

Expand both sides using the distributive property
Combine like terms
Subtract equivalent terms to isolate $ x $
Recognize contradictions indicating no solution

Final Answer

There is no solution to the equation $ 4(x - 3) + 2 = 2(2x + 1) - 6 $. The equation simplifies to a false statement, confirming the expression has no valid $ x $.

Bonus Tip: Applications of Solving Linear Equations

Mastering equations like this helps in real-life applications—from budgeting and physics to computer science. The algebraic skills developed here provide a strong foundation for more complex mathematical challenges.

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