Identifying The Mistake In Solving A Linear Equation A Step By Step Guide

Jul 10, 2025 by ADMIN 74 views

Identifying the Error in Solving a Linear Equation: A Step-by-Step Guide

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Linear equations are a fundamental concept in mathematics, serving as the building blocks for more advanced topics. Accurately solving these equations is crucial, but mistakes can easily occur if the correct steps and properties are not followed. This article delves into a specific example of solving a linear equation, highlighting a common error made during the process. We will break down each step, pinpoint the mistake, and reinforce the correct method for solving similar equations. Understanding these principles will not only improve your problem-solving skills but also build a solid foundation for tackling more complex mathematical challenges.

The Problem: A Step-by-Step Breakdown

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The equation we are examining is:

$\begin{aligned} 6 x-1 & =-2 x+9 \end{aligned}$

The presented solution attempts to solve for x through a series of algebraic manipulations. Let's dissect each step to identify the potential misstep.

The initial equation is: $6x - 1 = -2x + 9$ . This is a typical linear equation where our goal is to isolate the variable x on one side of the equation.

Step 1: Combining x terms

The solution proceeds to the next line as follows:

$\begin{aligned} 8 x-1 & =9 \end{aligned}$

To achieve this, the solver likely attempted to add $2x$ to both sides of the equation. The addition property of equality allows us to add the same value to both sides of an equation without changing its solution. Adding $2x$ to both sides of the original equation $6x - 1 = -2x + 9$ should result in:

$6x - 1 + 2x = -2x + 9 + 2x$

This simplifies to:

$8x - 1 = 9$

So far, the step is correct. This step successfully combines the x terms on the left side of the equation.

Step 2: Isolating the x term

The next step in the provided solution is:

$\begin{aligned} 8 x & =10 \end{aligned}$

Here, the solver seems to have added 1 to both sides of the equation $8x - 1 = 9$ . Again, the addition property of equality is applied. Adding 1 to both sides should give us:

$8x - 1 + 1 = 9 + 1$

Which simplifies to:

$8x = 10$

This step is also correct. We have successfully isolated the term with x on the left side.

Step 3: Solving for x

The next step is where the mistake occurs. The solution presents:

$\begin{aligned} x & = rac{8}{10} \end{aligned}$

To isolate x, we need to divide both sides of the equation $8x = 10$ by 8. The division property of equality states that we can divide both sides of an equation by the same non-zero value without changing the solution. Therefore, we should have:

$\frac{8x}{8} = \frac{10}{8}$

Which simplifies to:

$x = \frac{10}{8}$

The presented solution incorrectly states $x = \frac{8}{10}$ . This is a clear error in applying the division property of equality. The solver seems to have inverted the fraction, placing the coefficient of x (which is 8) in the numerator instead of the denominator.

Step 4: Simplifying the fraction

The final step in the provided solution is:

$\begin{aligned} x & =\frac{4}{5} \end{aligned}$

This step simplifies the incorrect fraction $\frac{8}{10}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2. While the simplification is done correctly, it is based on the incorrect value of x obtained in the previous step. The correct simplification should have been applied to $\frac{10}{8}$ .

The Mistake Pinpointed: Division Property of Equality

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The error lies in Step 3, where the division property of equality was incorrectly applied. Instead of dividing 10 by 8, the solver divided 8 by 10. This seemingly small mistake significantly alters the final result. Understanding the correct application of the division property is crucial for accurately solving linear equations.

Correcting the Solution: A Step-by-Step Approach

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To solidify understanding, let's solve the equation correctly, emphasizing the correct application of each property.

Original Equation:

$6x - 1 = -2x + 9$

Step 1: Add $2x$ to both sides (Addition Property of Equality)

$6x - 1 + 2x = -2x + 9 + 2x$

$8x - 1 = 9$

Step 2: Add 1 to both sides (Addition Property of Equality)

$8x - 1 + 1 = 9 + 1$

$8x = 10$

Step 3: Divide both sides by 8 (Division Property of Equality)

$\frac{8x}{8} = \frac{10}{8}$

$x = \frac{10}{8}$

Step 4: Simplify the fraction

$x = \frac{10 ÷ 2}{8 ÷ 2}$

$x = \frac{5}{4}$

Therefore, the correct solution to the equation $6x - 1 = -2x + 9$ is $x = \frac{5}{4}$ or $x = 1.25$ .

Why Understanding Properties of Equality is Crucial

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The properties of equality are the fundamental rules that govern how we manipulate equations in algebra. These properties ensure that we maintain the balance of the equation while isolating the variable we are solving for. A thorough understanding of these properties is essential for accurate problem-solving.

Key Properties Used:

Addition Property of Equality: If $a = b$ , then $a + c = b + c$ for any real number c. This property allows us to add the same value to both sides of an equation.
Subtraction Property of Equality: If $a = b$ , then $a - c = b - c$ for any real number c. This property allows us to subtract the same value from both sides of an equation.
Multiplication Property of Equality: If $a = b$ , then $ac = bc$ for any real number c. This property allows us to multiply both sides of an equation by the same value.
Division Property of Equality: If $a = b$ and $c ≠ 0$ , then $\frac{a}{c} = \frac{b}{c}$ . This property allows us to divide both sides of an equation by the same non-zero value.

Mistakes in applying these properties, as seen in the original solution, can lead to incorrect answers. Therefore, paying close attention to each step and ensuring the correct property is used is paramount.

Common Mistakes in Solving Linear Equations

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Besides the incorrect application of the division property of equality, several other common mistakes can occur when solving linear equations. Being aware of these pitfalls can help prevent errors and improve accuracy.

Incorrectly Combining Like Terms: This often involves combining terms that are not actually like terms (e.g., adding $3x$ and $2$ ).
Sign Errors: Mistakes in handling negative signs are common, especially when distributing a negative sign across parentheses.
Forgetting to Perform Operations on Both Sides: It's crucial to remember that any operation performed on one side of the equation must also be performed on the other side to maintain balance.
Incorrect Distribution: When dealing with expressions like $a(b + c)$ , the distributive property must be applied correctly, ensuring that a is multiplied by both b and c.

By carefully reviewing each step and double-checking for these common mistakes, you can significantly improve your accuracy in solving linear equations.

Practice Makes Perfect: Strengthening Your Skills

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Like any mathematical skill, solving linear equations requires practice. The more you practice, the more comfortable and confident you will become in applying the properties of equality and avoiding common mistakes. Here are some tips for effective practice:

Work through a variety of problems: Start with simpler equations and gradually progress to more complex ones.
Show your work: Writing down each step helps you identify potential errors and understand the process more clearly.
Check your answers: Substitute your solution back into the original equation to verify that it is correct.
Seek help when needed: Don't hesitate to ask your teacher, tutor, or classmates for assistance if you are struggling with a particular concept or problem.

By consistently practicing and reinforcing your understanding of the underlying principles, you can master the art of solving linear equations and build a strong foundation for future mathematical endeavors.

Conclusion: Mastering Linear Equations

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Solving linear equations is a fundamental skill in mathematics, and mastering it requires a clear understanding of the properties of equality and attention to detail. This article highlighted a common mistake in solving a linear equation – the incorrect application of the division property of equality. By carefully analyzing each step of the solution and identifying the error, we reinforced the correct method for solving such equations.

Remember, the key to success in mathematics lies in understanding the underlying concepts, practicing consistently, and learning from mistakes. By following the guidelines outlined in this article and diligently practicing, you can develop the skills and confidence needed to tackle any linear equation that comes your way. Always prioritize accuracy and a deep understanding of the principles involved. With consistent effort, you can master linear equations and pave the way for success in more advanced mathematical topics.