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:

Solved Solve the equation: | Chegg.com

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Solved Solve the equation: | Chegg.com
Solved 2/x-1 + 5/x+3 = 2 2(x+3) + 5(x-1) = 2(x+3)(x-1) 2x+6 | Chegg.com
Solved Solve the exact equation | Chegg.com
5) Solve for x in the equation 4 x - \frac { 3 x + 1 } { 2 } - \frac { 7
Question 5 - Solve the equation 2x + 1 = x - 3 - Chapter 4

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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Substituting \(n = 7\) and \(r = 4\), we get:
\binom{7}{4} = \frac{7!}{4! \times (7-4)!} = \frac{7 \times 6 \times 5 \times 4!}{4! \times 3 \times 2 \times 1} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35
Thus, there are \(\boxed{35}\) different sets of 4 instruments that can be chosen from the 7.

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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.