Solving Linear Equations A Step-by-Step Guide For X

In the realm of mathematics, solving for variables is a fundamental skill. One common type of equation encountered is the linear equation, where the highest power of the variable is 1. This article will provide a detailed, step-by-step guide on how to solve the linear equation:

$\frac{1}{3} x+\frac{1}{2}=-1(\frac{2}{3} x-2)$

Understanding the process of solving such equations is crucial for various mathematical applications and problem-solving scenarios. Before diving into the solution, let's break down the key concepts involved.

Understanding Linear Equations

At its core, a linear equation represents a straight line when graphed on a coordinate plane. The general form of a linear equation is:

$ax + b = c$

where 'x' is the variable, and 'a', 'b', and 'c' are constants. Solving for 'x' means isolating it on one side of the equation to determine its value. This value represents the point where the line intersects the x-axis, also known as the root or solution of the equation.

Key Principles for Solving Linear Equations:

To effectively solve linear equations, we rely on a few core principles that ensure we maintain the equality of the equation throughout the process:

1. Addition Property of Equality: Adding the same value to both sides of the equation does not change the solution.
2. Subtraction Property of Equality: Subtracting the same value from both sides of the equation does not change the solution.
3. Multiplication Property of Equality: Multiplying both sides of the equation by the same non-zero value does not change the solution.
4. Division Property of Equality: Dividing both sides of the equation by the same non-zero value does not change the solution.

These properties are the foundation of our step-by-step approach. We will use them strategically to manipulate the equation and isolate the variable 'x'.

Step-by-Step Solution

Now, let's apply these principles to solve the given equation:

$\frac{1}{3} x+\frac{1}{2}=-1(\frac{2}{3} x-2)$

Step 1: Distribute on the right side

The first step is to simplify the equation by distributing the -1 on the right side:

$\frac{1}{3} x+\frac{1}{2} = -1 * \frac{2}{3}x -1 * (-2)$

This simplifies to:

$\frac{1}{3} x+\frac{1}{2} = -\frac{2}{3}x + 2$

Step 2: Eliminate fractions by multiplying by the Least Common Multiple (LCM)

Fractions can make the equation look more complex. To get rid of them, we can multiply both sides of the equation by the Least Common Multiple (LCM) of the denominators. In this case, the denominators are 3 and 2, so the LCM is 6. Multiplying both sides by 6, we get:

$6 * (\frac{1}{3} x+\frac{1}{2}) = 6 * (-\frac{2}{3}x + 2)$

Distributing the 6 on both sides:

$6 * \frac{1}{3}x + 6 * \frac{1}{2} = 6 * (-\frac{2}{3}x) + 6 * 2$

This simplifies to:

$2x + 3 = -4x + 12$

Step 3: Isolate the variable terms on one side

Next, we want to gather all the terms containing 'x' on one side of the equation. We can do this by adding 4x to both sides:

$2x + 3 + 4x = -4x + 12 + 4x$

This simplifies to:

$6x + 3 = 12$

Step 4: Isolate the constant terms on the other side

Now, let's move the constant terms to the other side of the equation by subtracting 3 from both sides:

$6x + 3 - 3 = 12 - 3$

This simplifies to:

$6x = 9$

Step 5: Solve for x by dividing

Finally, to isolate 'x', we divide both sides of the equation by 6:

$\frac{6x}{6} = \frac{9}{6}$

This simplifies to:

$x = \frac{3}{2}$

Therefore, the solution to the equation is:

$x = \frac{3}{2}$ or $x = 1.5$

Verification

It's always a good idea to verify our solution by plugging it back into the original equation. Let's substitute $x = \frac{3}{2}$ into the original equation:

$\frac{1}{3} * \frac{3}{2} + \frac{1}{2} = -1(\frac{2}{3} * \frac{3}{2} - 2)$

Simplifying the left side:

$\frac{1}{2} + \frac{1}{2} = 1$

Simplifying the right side:

$-1(1 - 2) = -1(-1) = 1$

Since both sides are equal, our solution $x = \frac{3}{2}$ is correct.

Alternative Methods and Tips

While the step-by-step method provides a clear path to solving linear equations, there are alternative approaches and tips that can be helpful:

• Combining Like Terms: Before proceeding with the steps above, always look for opportunities to combine like terms on each side of the equation. This can simplify the equation and make it easier to solve.
• Alternative Fraction Elimination: Instead of multiplying by the LCM, you could multiply both sides by each denominator individually, one at a time. While this takes more steps, it can be more manageable for some.
• Mental Math: As you gain practice, you may find that you can perform some steps mentally, reducing the amount of written work.
• Checking for Extraneous Solutions: While not applicable in this specific linear equation, it's important to remember that when dealing with equations involving radicals or rational expressions, you must check for extraneous solutions. These are solutions that satisfy the transformed equation but not the original equation.

Common Mistakes to Avoid

When solving linear equations, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them:

• Incorrect Distribution: Ensure you distribute the multiplier to every term inside the parentheses.
• Combining Unlike Terms: Only combine terms that have the same variable and exponent.
• Sign Errors: Pay close attention to signs, especially when distributing negative numbers or moving terms across the equals sign.
• Incorrect Order of Operations: Follow the order of operations (PEMDAS/BODMAS) consistently.
• Forgetting to Multiply/Divide All Terms: When multiplying or dividing both sides of the equation, make sure to apply the operation to every term.

Real-World Applications of Solving Linear Equations

Linear equations aren't just abstract mathematical concepts; they have numerous real-world applications. Understanding how to solve them can be invaluable in various fields:

• Finance: Calculating interest, loan payments, and investments.
• Physics: Modeling motion, forces, and energy.
• Engineering: Designing structures, circuits, and systems.
• Chemistry: Balancing chemical equations and calculating concentrations.
• Everyday Life: Calculating expenses, budgeting, and making decisions based on numerical data.

Conclusion

In conclusion, mastering the art of solving linear equations is a crucial step in building a strong foundation in mathematics. By understanding the underlying principles, following a systematic approach, and practicing consistently, you can confidently tackle a wide range of linear equations. Remember to verify your solutions and be mindful of common mistakes. With dedication and the insights provided in this guide, you'll be well-equipped to solve for 'x' in any linear equation you encounter.

The solution to the equation $\frac{1}{3} x+\frac{1}{2}=-1(\frac{2}{3} x-2)$ is $x = \frac{3}{2}$ or $x = 1.5$. This guide has provided a comprehensive, step-by-step approach to solving linear equations, empowering you to tackle similar problems with confidence.