Interpreting Lily's Solution: A Math Equation Breakdown

by Alex Johnson 56 views

In this article, we will delve into the interpretation of a partial solution to a linear equation. Specifically, we'll examine Lily's work in solving the equation 4(x-1)-x=3(x+5)-11 and discuss how her steps lead to a particular conclusion about the equation's solution. Understanding these steps is crucial for grasping the fundamentals of algebra and problem-solving in mathematics.

Lily's Equation-Solving Journey

Let's start by revisiting the equation Lily is trying to solve:

4(x-1)-x=3(x+5)-11

Her initial steps are as follows:

  1. Expansion: Distribute the numbers outside the parentheses.
  2. Simplification: Combine like terms on each side of the equation.

Lily's work is presented as:

4(x-1)-x=3(x+5)-11
4x-4-x=3x+15-11
3x-4=3x+4

Step-by-Step Breakdown

Let’s break down each step to ensure we understand how Lily arrived at her partial solution.

  • Original Equation:

    4(x-1)-x=3(x+5)-11

This is the starting point. Lily's goal is to find the value(s) of x that make this equation true.

  • Expansion (Distributive Property):

    4x - 4 - x = 3x + 15 - 11

    Here, Lily applies the distributive property to expand both sides of the equation:

    • On the left side, 4 is multiplied by both x and -1.
    • On the right side, 3 is multiplied by both x and 5.

  • Simplification (Combining Like Terms):

    3x - 4 = 3x + 4

    In this step, Lily combines like terms on each side:

    • On the left side, 4x and -x are combined to get 3x.
    • On the right side, 15 and -11 are combined to get 4.

Interpreting the Partial Solution

Now, let's focus on the crucial part: interpreting the equation 3x - 4 = 3x + 4. This is where Lily's work provides a significant insight into the nature of the equation's solution.

Analyzing the Equation

The equation 3x - 4 = 3x + 4 presents a unique situation. Notice that we have the same x term (3x) on both sides. If we attempt to isolate x, we'll quickly see what happens.

Attempting to Isolate x

Let's try to subtract 3x from both sides of the equation:

3x - 4 - 3x = 3x + 4 - 3x

This simplifies to:

-4 = 4

This resulting statement, -4 = 4, is clearly false. This is a contradiction, indicating that there is no value of x that can make the original equation true.

The Significance of a Contradiction

The fact that we arrived at a contradiction (-4 = 4) tells us something very important about the equation 4(x-1)-x=3(x+5)-11. It means that the equation has no solution. In other words, there is no value for x that will satisfy the equation.

Understanding the Concept of No Solution

In linear equations, there are three possible outcomes when solving for x:

  1. One Unique Solution: This is the most common scenario, where you find a single value for x that makes the equation true (e.g., x = 2).
  2. Infinitely Many Solutions: This occurs when the equation simplifies to an identity, a statement that is always true (e.g., 0 = 0). In this case, any value of x will satisfy the equation.
  3. No Solution: This is what we encountered in Lily's equation. The equation simplifies to a contradiction, indicating that no value of x can make the equation true.

Why Did This Happen?

The reason Lily's equation has no solution lies in the structure of the equation itself. The coefficients of x on both sides are the same (both are 3), but the constant terms are different (-4 and 4). This creates an imbalance that cannot be resolved, regardless of the value of x.

Visualizing the Equation

To further illustrate this, consider graphing the two expressions on either side of the equation as two separate lines:

  • y = 3x - 4
  • y = 3x + 4

You would find that these lines are parallel. Parallel lines have the same slope (in this case, 3) but different y-intercepts (-4 and 4). Since they are parallel, they never intersect. The intersection point of two lines represents the solution to the equation formed by setting their expressions equal to each other. Since these lines don't intersect, there is no solution.

Common Mistakes to Avoid

When solving linear equations, it's easy to make mistakes. Here are a few common pitfalls to watch out for:

  1. Incorrect Distribution: Make sure to distribute correctly when expanding expressions. For example, ensure that you multiply the number outside the parentheses by every term inside the parentheses.
  2. Combining Unlike Terms: Only combine terms that are alike. For example, you can combine 3x and 2x, but you cannot combine 3x and 4.
  3. Incorrectly Applying Operations: Remember to perform the same operation on both sides of the equation to maintain balance. For example, if you subtract 4 from one side, you must subtract 4 from the other side as well.
  4. Misinterpreting Results: Be careful when interpreting the final result. A contradiction indicates no solution, while an identity indicates infinitely many solutions.

Conclusion

In summary, Lily's partial solution to the equation 4(x-1)-x=3(x+5)-11, which led to the contradiction -4 = 4, indicates that the equation has no solution. This understanding is a fundamental aspect of solving linear equations and interpreting their results. Recognizing contradictions and identities is key to accurately determining the nature of the solution set.

By carefully analyzing the steps in solving an equation and understanding the implications of the results, students can develop a strong foundation in algebra. Remember to pay attention to detail, avoid common mistakes, and always check your work.

For further exploration of linear equations and their solutions, consider visiting resources like Khan Academy's Linear Equations Section.