Solving Systems Of Equations: Understanding Infinite Solutions

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Hey guys! Let's break down this math problem. We're looking at a system of equations, and the goal is to figure out the solution. The work provided gives us some clues, so let's carefully analyze it to determine the correct answer. Systems of equations are sets of two or more equations that we try to solve together. The solution to a system of equations is the set of values that satisfy all equations in the system. There are a few possibilities: we can have one unique solution, no solution, or infinitely many solutions. In this particular case, we need to understand what the provided work tells us about the type of solution we have. Let's delve into the details and the reasoning behind each choice to find out what's going on.

Analyzing the Given Equations and Work

The system of equations presented to us is:

  • 5x + 2y = 8
  • -4(1.25x + 0.5y = 2)

First, let's simplify the second equation. Multiply the terms inside the parenthesis by -4:

  • -4 * 1.25x = -5x
  • -4 * 0.5y = -2y
  • -4 * 2 = -8

So, the second equation becomes -5x - 2y = -8.

Now, here's the clever part. We are then adding the two equations together. Adding the first equation (5x + 2y = 8) to the modified second equation (-5x - 2y = -8), we get:

  • 5x + (-5x) = 0
  • 2y + (-2y) = 0
  • 8 + (-8) = 0

The result is 0 = 0. This is where the magic happens! When we get a true statement like 0 = 0, it means that the two equations are essentially the same (or multiples of each other), which indicates that we have infinitely many solutions. This means there isn't just one specific point where the lines intersect; instead, the lines are overlapping, and every point on the line is a solution. This is a crucial concept to grasp when solving systems of equations. It shows us that not all systems of equations have a single, definitive answer. Sometimes, the relationship between the equations leads to a broader, more general solution.

Understanding the Implications of 0 = 0

When we arrive at a statement like 0 = 0 during the process of solving a system of equations, it tells us something profound about the relationship between the equations. This outcome does not represent a specific point of intersection like you'd get with a unique solution (e.g., x=2, y=3). Instead, it signifies that the two equations are dependent, which means that they represent the same line or are multiples of each other. Think of it like this: if you graph both equations, they would overlap completely, with every single point on that line being a solution. This situation contrasts sharply with a system that has no solution at all, where the lines are parallel and never intersect. It also contrasts with systems where the lines intersect at only one point, giving a unique solution. The 0 = 0 result is the signal that there are infinitely many solutions, all the points on the common line. Understanding this distinction is key to interpreting the result of solving systems of equations.

Decoding the Answer Choices and Finding the Correct Solution

Let's go through the answer choices step by step. We'll examine each option based on our analysis of the given equations and the resulting 0 = 0 statement. This process will help us select the right solution to the system.

The Answer Choices

Here are the options we have to choose from:

  • A. (-4, -4)
  • B. (0, 0)
  • C. no solution
  • D. infinitely many solutions

Analyzing the Options

  • A. (-4, -4): This suggests a single point solution. However, since the equations simplify to 0 = 0, this is not correct. If we plug these values into the original equations:

    • 5(-4) + 2(-4) = -20 - 8 = -28 ≠ 8
    • -4[1.25(-4) + 0.5(-4)] = -4[-5 - 2] = -4(-7) = 28 ≠ 8 This point does not satisfy either equation.

  • B. (0, 0): This, too, indicates a single point solution. Again, this is not correct.

    • 5(0) + 2(0) = 0 ≠ 8
    • -4[1.25(0) + 0.5(0)] = 0 ≠ 8 This point also does not satisfy either equation.

  • C. no solution: This would happen if the lines were parallel, meaning they never intersect. However, since we got 0 = 0, the lines are not parallel but are the same. Thus, this is incorrect.

  • D. infinitely many solutions: This is consistent with our simplification to 0 = 0. The equations are dependent and represent the same line. Every point on that line satisfies both equations. This is the correct answer.

Conclusion

Based on our step-by-step analysis, the correct answer is D. infinitely many solutions. The manipulation of the equations led us to 0 = 0, which indicates that the lines are coincident (the same) and that there is an infinite number of points that satisfy both equations. Nice job, guys! You've successfully navigated a system of equations problem!

Additional Tips for Solving Systems of Equations

Here are some extra tips to help you in your future endeavors with systems of equations. These pointers will help you become more comfortable and proficient in tackling these types of problems. Let's get to it!

Methods for Solving Systems of Equations

  • Substitution: Solve one equation for one variable and substitute that expression into the other equation. This method is especially useful when one of the equations is already solved for a variable or can easily be rearranged to do so. It helps simplify the system into a single equation with one variable.
  • Elimination: Add or subtract the equations to eliminate one of the variables. This method is effective when the coefficients of one of the variables are opposites or can be easily made so. Multiplying one or both equations by a constant can help you set up the elimination process.
  • Graphing: Graph both equations on the same coordinate plane. The point(s) of intersection represent the solution(s) to the system. This method is visual and helps you understand the nature of the solutions, but it may not always be the most precise, especially if the intersection points have non-integer coordinates.

Recognizing Special Cases

  • No Solution: If, after solving the system, you get a false statement like 2 = 5, then the system has no solution. This indicates that the lines are parallel and never intersect.
  • Infinitely Many Solutions: If, after solving the system, you get a true statement like 0 = 0 or x = x, then the system has infinitely many solutions. This means the equations represent the same line.

Practicing and Understanding

  • Practice, Practice, Practice: The more problems you solve, the more comfortable you will become with different types of systems and solution methods. Work through various examples, starting with simpler systems and gradually increasing the complexity.
  • Understand the Concepts: Focus on grasping the underlying concepts rather than just memorizing steps. Understand why different methods work and when they are most appropriate.
  • Check Your Answers: Always verify your solutions by substituting the values back into the original equations. This helps ensure that your solution is correct and that you haven't made any computational errors.

Resources for Further Learning

  • Khan Academy: Offers free video tutorials, practice exercises, and articles on systems of equations and related topics. A great place for beginners to start.
  • Mathway: A helpful online tool where you can input systems of equations and get step-by-step solutions. Useful for checking your work and understanding the solving process.
  • Textbooks and Workbooks: Traditional textbooks and workbooks provide a structured approach to learning and practicing systems of equations. Look for ones with plenty of examples and practice problems.

By following these tips and utilizing the provided resources, you'll be well on your way to mastering systems of equations. Keep practicing, stay curious, and you'll find that solving these problems becomes easier and more enjoyable over time! Keep up the great work! You got this!