Consistent & Independent Systems: Equation Pairing Guide

by Alex Johnson 57 views

Understanding systems of equations can feel like navigating a maze, especially when terms like "consistent" and "independent" come into play. If you're grappling with the question, "Which equation can pair with 3x + 4y = 8 to create a consistent and independent system?", you've come to the right place. Let's break down the concepts and solve this problem step by step, making it crystal clear for you.

Decoding Consistent and Independent Systems

Before diving into the specific equation, it's crucial to understand what a "consistent and independent system" actually means. In the realm of linear equations, we often deal with pairs of equations, and their solutions represent the points where the lines intersect on a graph. The nature of this intersection (or lack thereof) determines the system's classification:

  • Consistent System: A consistent system is one that has at least one solution. This means the lines either intersect at a single point or are the same line (intersecting at infinite points).
  • Inconsistent System: An inconsistent system has no solution. This occurs when the lines are parallel and never intersect.
  • Independent System: An independent system has exactly one solution. The lines intersect at a single, unique point.
  • Dependent System: A dependent system has infinitely many solutions. This happens when the two equations represent the same line.

So, a consistent and independent system is a pair of linear equations that intersect at exactly one point. This is the key to solving our problem.

Analyzing the Given Equation: 3x + 4y = 8

We're given the equation 3x + 4y = 8. To find another equation that forms a consistent and independent system with this one, we need an equation that will intersect this line at only one point. This means the lines cannot be parallel (inconsistent) and cannot be the same line (dependent). Graphically, two lines intersect at a single point if they have different slopes.

The given equation is in standard form (Ax + By = C). We can rewrite it in slope-intercept form (y = mx + b) to easily identify its slope:

4y = -3x + 8 y = (-3/4)x + 2

Thus, the slope of our given line is -3/4. Any line that forms a consistent and independent system with this line must have a different slope.

Evaluating the Answer Choices

Now, let's examine the provided answer choices and determine which equation has a slope different from -3/4:

A. 6x + 8y = 16 B. -3x - 4y = -6 C. 6x - 3y = 2 D. -3x - 4y = -8

We'll convert each equation to slope-intercept form (y = mx + b) to find their slopes.

Choice A: 6x + 8y = 16

8y = -6x + 16 y = (-6/8)x + 2 y = (-3/4)x + 2

The slope of this line is -3/4, which is the same as our given line. Furthermore, if we divide the entire equation by 2, we get 3x + 4y = 8, which is exactly the same equation. This means the lines are identical, and the system is dependent (infinitely many solutions), not independent.

Choice B: -3x - 4y = -6

-4y = 3x - 6 y = (-3/4)x + 3/2

The slope of this line is also -3/4. Since the slopes are the same, the lines are either parallel or the same line. However, the y-intercept (3/2) is different from the y-intercept of our given line (2). Therefore, these lines are parallel and do not intersect. This system is inconsistent (no solution), not consistent and independent.

Choice C: 6x - 3y = 2

-3y = -6x + 2 y = 2x - 2/3

The slope of this line is 2, which is different from -3/4. Since the slopes are different, the lines will intersect at a single point, forming a consistent and independent system. This is the correct choice.

Choice D: -3x - 4y = -8

-4y = 3x - 8 y = (-3/4)x + 2

Again, the slope is -3/4, which is the same as our original line. Also, we can see that the y-intercept is 2, so this line is the same as the original one. This represents a dependent system with infinitely many solutions, not an independent one.

Conclusion: The Correct Pairing Equation

After analyzing all the choices, we've determined that the equation 6x - 3y = 2 (Choice C) is the only one that, when paired with 3x + 4y = 8, creates a consistent and independent system. This is because it has a different slope, ensuring the lines intersect at exactly one point.

Key Takeaways

  • A consistent and independent system has exactly one solution, meaning the lines intersect at a single point.
  • To form a consistent and independent system with a given line, the new line must have a different slope.
  • Converting equations to slope-intercept form (y = mx + b) makes it easy to identify the slope (m) and y-intercept (b).
  • Parallel lines have the same slope but different y-intercepts.
  • The same line has the same slope and the same y-intercept.

By understanding these concepts, you can confidently tackle similar problems involving systems of equations. Remember to focus on the slopes and y-intercepts to quickly determine the nature of the system.

For further exploration of linear equations and systems, consider visiting Khan Academy's Linear Equations and Inequalities Section, a trusted resource for mathematics education.