Marta's Mistake Solving Equations A Mathematical Puzzle

In the realm of mathematics, solving systems of equations is a fundamental skill. It's like detective work, where we use clues (equations) to uncover the values of unknown variables. Marta attempted to solve a system of equations, but unfortunately, she stumbled upon a mistake. Let's delve into the problem, dissect her steps, and pinpoint where she went wrong. Understanding these errors is crucial for anyone learning algebra, as it highlights common pitfalls and reinforces the importance of careful manipulation of equations. By the end of this exploration, you'll not only understand Marta's mistake but also gain a deeper understanding of how to solve such systems accurately.

The System of Equations

The system of equations Marta tried to solve is:

10x + y = 22
2x + y = -2

This system presents two linear equations, each containing two variables, x and y. The goal is to find the values of x and y that satisfy both equations simultaneously. There are several methods to tackle this kind of problem, including substitution, elimination, and graphing. Marta opted for a substitution approach, which involves solving one equation for one variable and then substituting that expression into the other equation. This method, when applied correctly, can systematically reduce the problem to a single-variable equation, making it easier to solve. However, it's also a method where errors can easily creep in if not handled with precision and care. The beauty of solving systems of equations lies in the variety of approaches available, each with its own strengths and weaknesses, and the satisfaction of arriving at the correct solution. Let's see where Marta's approach led her astray.

Marta's Attempt

Marta's steps are as follows:

1. y = 22 - 10x (Solved the first equation for y)
2. 10x + (22 - 10x) = 22 (Substituted the expression for y into the first equation)
3. 10x + 22 - 10x = 22

Marta started by isolating y in the first equation, which is a standard and valid initial move in the substitution method. She correctly rearranged the equation 10x + y = 22 to get y = 22 - 10x. This step demonstrates a good understanding of algebraic manipulation. The next logical step is to substitute this expression for y into the second equation, not the first one. This is where the critical mistake occurs. Substituting back into the same equation she used to solve for y doesn't provide any new information and won't lead to a solution for x or y. It's akin to trying to solve a puzzle by only looking at one piece; you need to see how the pieces fit together. By substituting into the wrong equation, Marta essentially created a tautology, an equation that is always true but doesn't help in finding the specific values of x and y that satisfy the system. This highlights the importance of carefully tracking which equation you're working with and ensuring you're introducing new information into the system.

Identifying the Mistake

The critical mistake lies in substituting the expression for y back into the same equation from which it was derived. Marta solved the first equation for y and then substituted that expression back into the first equation. This process doesn't eliminate any variables or provide new information; it simply confirms the original equation's structure. To correctly apply the substitution method, the expression for y should have been substituted into the second equation (2x + y = -2). This substitution would have created an equation with only x as the variable, allowing Marta to solve for x. Identifying this error is a key learning point for anyone studying algebra. It underscores the need to understand the purpose of each step in the solution process, not just the mechanics. The goal of substitution is to reduce the system to a single equation with a single variable, and substituting back into the original equation defeats this purpose.

The Correct Approach

To correctly solve the system, we should substitute y = 22 - 10x into the second equation:

2x + y = -2
2x + (22 - 10x) = -2

Now we have an equation with only x as the variable. Let's simplify and solve for x:

2x + 22 - 10x = -2
-8x + 22 = -2
-8x = -24
x = 3

With the value of x now known, we can substitute it back into either of the original equations to find y. Let's use the first equation:

10x + y = 22
10(3) + y = 22
30 + y = 22
y = -8

Therefore, the solution to the system of equations is x = 3 and y = -8. This correct approach demonstrates the power of substitution when applied strategically. By substituting into the other equation, we successfully eliminated a variable and created a solvable equation. This highlights the importance of understanding the underlying principles of algebraic techniques and applying them with precision. The solution, x = 3 and y = -8, satisfies both original equations, confirming its accuracy.

Why This Mistake Matters

Marta's mistake is a common one in algebra, and understanding why it's incorrect is crucial for mastering equation-solving techniques. By substituting back into the same equation, no new information is introduced, and the system doesn't get simplified. It's like trying to unlock a door with the same key that opened the first lock in a series; you need a different key for the next lock. This error underscores the fundamental principle of equation solving: each step should bring you closer to isolating the variables and finding their values. Substituting back into the original equation essentially spins your wheels, leaving you where you started. Recognizing this type of mistake not only improves your algebraic skills but also enhances your problem-solving abilities in other areas. It teaches you to be methodical, to track your steps, and to always ask yourself if your actions are moving you closer to the solution. Mathematics, at its core, is about logical progression, and avoiding pitfalls like Marta's mistake is key to that progression.

Key Takeaways

• Always substitute into the other equation: When using the substitution method, the expression for a variable should be substituted into the equation that was not used to derive that expression.
• Understand the goal: The purpose of substitution is to eliminate a variable and create a simpler equation.
• Check your work: After finding a solution, substitute the values back into the original equations to verify they are satisfied.

By understanding Marta's mistake and the correct approach, you can strengthen your problem-solving skills and avoid similar errors in the future. Solving systems of equations is a critical skill in algebra and beyond, and mastering it requires both knowledge of the techniques and careful attention to detail. Remember, mathematics is not just about getting the right answer; it's about understanding the process and the reasoning behind each step. By analyzing errors like Marta's, we gain valuable insights into the nuances of algebraic manipulation and develop a deeper appreciation for the elegance and precision of mathematics.

What was Marta's mistake in solving the system of equations? Explain the correct method to solve the system.

Marta's Mistake in Solving Equations A Mathematical Puzzle