Micah's Equation: Unraveling The Steps To The Solution

by Alex Johnson 55 views

Let's dive into Micah's work on solving a linear equation and see what we can learn! Micah tackled the equation and arrived at a solution, $x=0$. We'll examine his steps, pinpoint any potential areas for improvement, and discuss the core concepts of solving linear equations. Understanding how equations work is like having a secret decoder ring for the world of math – it unlocks the ability to solve all sorts of problems! So, get ready to sharpen your equation-solving skills as we explore Micah's method.

Breaking Down Micah's Linear Equation

Micah started with the equation $ rac{5}{6}(1-3 x) =4ig(- rac{5 x}{8}+2ig)$. The goal is to isolate x and find its value. His initial approach involved simplifying the equation by distributing the constants on both sides. This is a crucial step in solving linear equations, as it removes parentheses and allows for the terms to be rearranged. It's like unwrapping a present – you need to get rid of the wrapping paper (parentheses) to see what's inside. Let's analyze his distribution steps to check whether they are correct and easy to follow. A mistake in the distribution process can lead to the wrong answer, so paying close attention to these steps is extremely important. We can then compare the results to see if everything is correct.

Firstly, Micah distributed rac{5}{6} across the terms inside the parentheses on the left side, which is (1−3x)(1 - 3x). This multiplication resulted in rac{5}{6} imes 1 = rac{5}{6} and rac{5}{6} imes (-3x) = - rac{5x}{2}. Therefore, the left side of the equation became rac{5}{6} - rac{5x}{2}. This process is the correct application of the distributive property of multiplication over subtraction. He then moved on to the right side of the equation, which is 4(- rac{5x}{8} + 2). Here, Micah distributed the 44 across the terms inside the parentheses. So, he calculated 4 imes - rac{5x}{8} = - rac{5x}{2} and 4imes2=84 imes 2 = 8. This gives - rac{5x}{2} + 8. Therefore, the equation becomes rac{5}{6} - rac{5x}{2} = - rac{5x}{2} + 8.

The Distributive Property

The distributive property is a fundamental concept in algebra. In essence, it tells us how to multiply a number by a sum or difference inside parentheses. For example, a(b+c)=ab+aca(b + c) = ab + ac. It's like saying you can either give each person a gift individually or give the entire pile of gifts to each person. In Micah's case, he correctly applied this property to both sides of the equation. This simplification step is important because it makes the equation simpler to work with, bringing us closer to isolating the variable x. It is especially helpful when dealing with equations that have parentheses or require us to combine multiple terms. Mastering this property is essential for students learning to solve equations.

Examining Micah's Subsequent Steps

After distributing, Micah had the equation rac{5}{6} - rac{5x}{2} = - rac{5x}{2} + 8. He seems to have noticed that he had - rac{5x}{2} on both sides of the equation. Micah appears to have moved terms around in an attempt to simplify and solve for x. The subsequent step should involve isolating the variable x on one side of the equation. To do this, Micah could have tried to eliminate terms by adding or subtracting the same values from both sides.

However, in his final step, Micah writes 0=80 = 8. This is clearly not correct. Let's trace the steps and uncover the source of the error. When we look closely at Micah's equation after distribution, we can see that the - rac{5x}{2} term appears on both sides. If Micah adds rac{5x}{2} to both sides, the x term would disappear from both sides. This would leave Micah with rac{5}{6} = 8. This is a false statement. Since the x term vanishes, it implies that the equation has either no solution or infinitely many solutions. In this case, since rac{5}{6} does not equal 8, it means that the equation has no solution. Micah seems to have overlooked the fact that, in this case, x does not actually have a value. Therefore, it is important to remember that not all equations have solutions, and the correct approach is essential when solving these equations.

Potential Errors and Considerations

One potential error could have been in the initial distribution process, although the steps seem correct at a glance. Another area to check is the handling of the variable x. It is possible that a sign error or miscalculation could have led to this incorrect conclusion. To solve a linear equation, the goal is always to isolate the variable on one side of the equation. This involves a series of steps that includes the distributive property, combining like terms, and using inverse operations (addition/subtraction, multiplication/division) to undo the operations performed on the variable.

The presence of the same x term on both sides suggests that x might not have a unique solution or there might be no solution. Micah needs to carefully review his steps to make sure he has not made any errors in his calculations. The key is to carefully track each step and double-check the calculations.

Conclusion: Where Did Micah Go Wrong?

Micah's final step, resulting in 0=80 = 8, indicates an issue with his understanding or execution of solving the equation. The equation should not have been solved by assuming that the solution is x=0x = 0. In this case, the equation had no solution, since the x terms canceled out and resulted in an inconsistent equation. This conclusion should have been drawn as a result. If Micah would have added rac{5x}{2} to both sides of the equation rac{5}{6} - rac{5x}{2} = - rac{5x}{2} + 8, he would be left with the equation rac{5}{6} = 8, which would tell him that the equation is false, and there is no solution. Micah needs to practice and get more familiar with the strategies to solve this kind of linear equation. Regular practice and seeking help when needed are key ingredients to master the skill of solving linear equations.

Tips for Solving Linear Equations

Here's a quick guide to help you conquer linear equations:

  • Simplify: Use the distributive property to eliminate parentheses. Combine like terms on each side.
  • Isolate: Get all the x terms on one side and the constants on the other using addition, subtraction, multiplication, or division.
  • Solve: Solve for x by performing the final operations to isolate it.
  • Check: Plug your solution back into the original equation to verify your answer.

By following these steps and understanding the underlying concepts, you can confidently solve any linear equation that comes your way. It might seem tricky at first, but with practice, it becomes second nature! Remember, mathematics is about logical thinking and the application of rules, so keep practicing and you will do great!

For further information, check out this Khan Academy guide on solving linear equations!