Question: Solve for $ x $ in the equation $ 4(x - 3) + 2 = 2(2x + 1) - 6 $ - Imagemakers
Solving Linear Equations: Step-by-Step Guide to Solve for $ x $
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:
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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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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.
