Solving Linear Equations A Step By Step Guide

In this article, we will delve into the process of solving for the variable x in a linear equation. Linear equations, which involve a variable raised to the power of one, are fundamental in mathematics and have wide-ranging applications in various fields. We will break down the steps involved in solving the equation:

$$\frac{1}{2} x+\frac{1}{4}=2\left(\frac{4}{5} x+4\right)$$

This equation may appear complex at first glance, but by systematically applying algebraic principles, we can isolate x and determine its value. Let's embark on this mathematical journey together!

1. Simplify the Equation: Eliminating Parentheses and Fractions

Our initial equation is:

$$\frac{1}{2} x+\frac{1}{4}=2\left(\frac{4}{5} x+4\right)$$

The first step towards simplifying this equation is to eliminate the parentheses. We achieve this by distributing the '2' on the right side of the equation:

$$\frac{1}{2} x+\frac{1}{4} = 2 * \frac{4}{5} x + 2 * 4$$

This simplifies to:

$$\frac{1}{2} x+\frac{1}{4} = \frac{8}{5} x + 8$$

Now, we need to tackle the fractions. Fractions can make equations appear more complicated, so eliminating them is a crucial simplification step. To do this, we find the least common multiple (LCM) of the denominators. The denominators in our equation are 2, 4, and 5. The LCM of these numbers is 20. We multiply both sides of the equation by 20:

$$20 * \left(\frac{1}{2} x+\frac{1}{4}\right) = 20 * \left(\frac{8}{5} x + 8\right)$$

Distributing the 20 on both sides, we get:

$$20 * \frac{1}{2} x + 20 * \frac{1}{4} = 20 * \frac{8}{5} x + 20 * 8$$

Simplifying each term:

$$10x + 5 = 32x + 160$$

By strategically eliminating parentheses and fractions, we have transformed the original equation into a much simpler form. This simplified equation is now easier to manipulate and solve for x. This step is crucial for ensuring accuracy and efficiency in the subsequent steps.

2. Isolate the Variable Term: Rearranging the Equation

Having simplified the equation to:

$$10x + 5 = 32x + 160$$

our next goal is to isolate the terms containing x on one side of the equation. This involves rearranging the equation by adding or subtracting terms from both sides. The key is to perform the same operation on both sides to maintain the equality.

Let's begin by subtracting 10x from both sides of the equation:

$$10x + 5 - 10x = 32x + 160 - 10x$$

This simplifies to:

$$5 = 22x + 160$$

Now, we need to isolate the term with x further. To do this, we subtract 160 from both sides of the equation:

$$5 - 160 = 22x + 160 - 160$$

This simplifies to:

$$-155 = 22x$$

By strategically adding and subtracting terms, we have successfully isolated the variable term (22x) on one side of the equation and a constant term (-155) on the other side. This isolation is a critical step in solving for x, as it brings us closer to determining its value.

3. Solve for x: Division to Isolate the Variable

At this stage, we have the equation:

$$-155 = 22x$$

Our final step is to isolate x completely. Currently, x is being multiplied by 22. To undo this multiplication, we perform the inverse operation: division. We divide both sides of the equation by 22:

$$\frac{-155}{22} = \frac{22x}{22}$$

This simplifies to:

$$x = -\frac{155}{22}$$

We have now successfully solved for x. The value of x that satisfies the original equation is -155/22. This fraction can be left as is, or it can be converted to a decimal approximation if desired. The key is that we have isolated x and determined its exact value.

4. Verification: Substituting the Solution Back into the Original Equation

To ensure the accuracy of our solution, it's essential to verify it. Verification involves substituting the value we found for x back into the original equation. If the equation holds true after the substitution, our solution is correct. The original equation was:

$$\frac{1}{2} x+\frac{1}{4}=2\left(\frac{4}{5} x+4\right)$$

We found that x = -155/22. Let's substitute this value into the equation:

$$\frac{1}{2} * \left(-\frac{155}{22}\right) + \frac{1}{4} = 2 * \left(\frac{4}{5} * \left(-\frac{155}{22}\right) + 4\right)$$

Let's simplify both sides of the equation separately. Starting with the left side:

$$\frac{1}{2} * \left(-\frac{155}{22}\right) + \frac{1}{4} = -\frac{155}{44} + \frac{1}{4}$$

To add these fractions, we need a common denominator, which is 44:

$$- \frac{155}{44} + \frac{1}{4} = -\frac{155}{44} + \frac{11}{44} = -\frac{144}{44}$$

Simplifying the fraction by dividing both numerator and denominator by 4:

$$- \frac{144}{44} = -\frac{36}{11}$$

Now, let's simplify the right side of the equation:

$$2 * \left(\frac{4}{5} * \left(-\frac{155}{22}\right) + 4\right) = 2 * \left(-\frac{4 * 155}{5 * 22} + 4\right)$$

Simplifying the fraction inside the parentheses:

$$2 * \left(-\frac{620}{110} + 4\right) = 2 * \left(-\frac{62}{11} + 4\right)$$

Converting 4 to a fraction with a denominator of 11:

$$2 * \left(-\frac{62}{11} + \frac{44}{11}\right) = 2 * \left(-\frac{18}{11}\right)$$

Multiplying by 2:

$$2 * \left(-\frac{18}{11}\right) = -\frac{36}{11}$$

Comparing both sides, we see that:

$$- \frac{36}{11} = -\frac{36}{11}$$

The left side equals the right side, confirming that our solution, x = -155/22, is correct. Verification is a crucial step in the problem-solving process. It provides confidence in the accuracy of the solution and helps identify any potential errors.

Conclusion: Mastering the Art of Solving Linear Equations

In this comprehensive guide, we have walked through the process of solving for x in the linear equation:

$$\frac{1}{2} x+\frac{1}{4}=2\left(\frac{4}{5} x+4\right)$$

We systematically applied algebraic principles, including simplifying the equation by eliminating parentheses and fractions, isolating the variable term, and dividing to solve for x. We also emphasized the importance of verification to ensure the accuracy of our solution. The solution we found is:

$$x = -\frac{155}{22}$$

Mastering the art of solving linear equations is a fundamental skill in mathematics. These equations form the basis for more advanced mathematical concepts and have applications in various fields, including science, engineering, and economics. By understanding the steps involved and practicing regularly, you can develop confidence and proficiency in solving linear equations.

This article provides a solid foundation for tackling linear equations. Remember to approach each equation systematically, break it down into manageable steps, and always verify your solution. With practice, you can become a proficient problem solver and confidently tackle more complex mathematical challenges.