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:

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

Understanding the Context

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:

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

Key Insights

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

Now simplify both sides:

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

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

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Question: A polynomial $ f(x) $ satisfies $ f(x + 1) - f(x) = 6x + 4 $. Find the leading coefficient of $ f(x) $.
Solution: Since the difference $ f(x+1) - f(x) $ is linear, $ f(x) $ must be a quadratic polynomial. Let $ f(x) = ax^2 + bx + c $. Then:
f(x+1) = a(x+1)^2 + b(x+1) + c = a(x^2 + 2x + 1) + b(x + 1) + c = ax^2 + 2ax + a + bx + b + c

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Final Thoughts

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.