Consider The Following System Of Equations Y 2x 2y 2x 7 What

Consider The Following System Of Equationsy2x 2y 2x7what Can

Consider the following system of equations: y = 2x - 2 and y - 2x = 7. The task is to analyze the system to determine whether it is consistent, inconsistent, dependent, or independent, and then find the solution(s) if any exist.

To analyze this system, note that the first equation is written in slope-intercept form, y = 2x - 2. The second equation, y - 2x = 7, can also be rearranged into the slope-intercept form: y = 2x + 7. Comparing these two equations, y = 2x - 2 and y = 2x + 7, reveals that both equations have the same slope, 2, but different y-intercepts (-2 and 7 respectively). This indicates that the two lines are parallel and will never intersect.

Since the lines are parallel and do not intersect, the system is inconsistent. Therefore, the correct conclusion is:

Paper For Above instruction

The system of equations given is:

y = 2x - 2

y - 2x = 7

Rearranging the second equation into slope-intercept form yields y = 2x + 7. Comparing both equations, the slopes (2) are identical, but the y-intercepts differ. Since parallel lines do not intersect, there are no solutions satisfying both equations simultaneously. This indicates that the system is inconsistent, having no common solution. It is not dependent (which would mean infinitely many solutions) nor independent with a unique solution. Therefore, the system of equations has no solution, confirming its inconsistency in the plane.

Analysis of Solutions

Given the above, the solution set is empty. To verify this, consider solving the system algebraically:

Substitute y = 2x - 2 into the second equation:


(2x - 2) - 2x = 7


2x - 2 - 2x = 7


-2 = 7

This contradiction confirms that the equations do not share a common point, thus no solution exists.

Implications and Conclusions

The key takeaway from this analysis is understanding the relationship between the equations’ slopes and intercepts. Parallel lines, characterized by equal slopes but different intercepts, eliminate the possibility of solutions. This is a fundamental concept in both algebra and analytic geometry, illustrating the importance of slope comparisons when analyzing systems of linear equations.

This concept directly applies to various real-world scenarios, such as in business planning, engineering, and logistics, where parallel or non-intersecting relationships can indicate constraints or lack of overlap between different processes or supply chains.

References

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  • Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
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  • Hirsch, M. W., & Smale, S. (2014). Differential Equations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
  • Strogatz, S. H. (2018). Nonlinear Dynamics and Chaos. CRC Press.
  • Ross, K. (2013). Elementary Linear Algebra. Pearson.
  • Gerard, W., & Wu, C. (2011). The Geometry of Lines and Planes. Mathematics in Engineering.
  • Hoffman, K., & Kunze, R. (2010). Linear Algebra. Prentice Hall.

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