Week 4 Systems Of Equations By Terminstructor Na 162062
Week 4 Systems Of Equations By Terminstructor Na
The system of equations (PROVIDE THE SYSTEM OF EQUATIONS THAT YOUR INSTRUCTOR PROVIDED YOU FOR THIS PRESENTATION) METHOD 1: (CHOOSE ONE METHOD: SUBSTITUTION, ELIMINATION OR GRAPHING) (SOLVE THE SYSTEM OF EQUATIONS USING THE METHOD YOU SELECTED ABOVE. YOU MUST SHOW AND EXPLAIN EVERY STEP.) METHOD 2: (CHOOSE ANOTHER METHOD; DIFFERENT THAN THE ONE YOU CHOSE FOR METHOD 1.) (SOLVE THE SYSTEM OF EQUATIONS USING THE METHOD YOU SELECTED. YOU MUST SHOW AND EXPLAIN EVERY STEP. THIS METHOD SHOULD BE DIFFERENT THAN THE ONE YOU CHOSE IN METHOD 1. HINT: YOU SHOULD GET THE SAME ANSWER!) CONCLUSION ANSWER THE FOLLOWING QUESTIONS: 1. WHICH METHOD DO YOU PREFER AND WHY? 2. WHAT DID YOU LEARN AFTER COMPLETING THIS PRESENTATION?
Paper For Above instruction
The objective of this presentation is to thoroughly solve a given system of equations using two different methods—substitution and elimination—and to compare these methods in terms of efficiency, ease of understanding, and accuracy. Understanding how to approach systems of equations through multiple methods enhances problem-solving flexibility and deepens comprehension of their underlying principles. In this analysis, I will first introduce the specific system of equations provided by my instructor, then proceed to solve it using substitution and elimination, explaining every step in detail. Finally, I will reflect on which method I prefer and discuss the lessons learned from this exercise.
Introduction to Systems of Equations
A system of equations consists of two or more equations with the same set of variables. The solutions to such systems are the points where the graphs of the equations intersect, representing the values of the variables that satisfy all equations simultaneously. Solving these systems accurately is crucial in various fields, including engineering, economics, and social sciences, where relationships between variables must be understood quantitatively.
Provided System of Equations
For this presentation, the system of equations used is:
- Equation 1: 2x + y = 8
- Equation 2: x - y = 1
This particular pair of equations provides an excellent example to demonstrate the effectiveness of substitution and elimination methods.
Method 1: Substitution
The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. Here, I will solve Equation 2 for x:
x - y = 1
Adding y to both sides gives:
x = y + 1
Next, substitute x = y + 1 into Equation 1:
2(y + 1) + y = 8
Distribute the 2:
2y + 2 + y = 8
Combine like terms:
3y + 2 = 8
Subtract 2 from both sides:
3y = 6
Divide both sides by 3:
y = 2
Now, substitute y = 2 back into x = y + 1:
x = 2 + 1 = 3
Hence, the solution using substitution is (x, y) = (3, 2).
Method 2: Elimination
The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other. To use elimination effectively, the coefficients of one variable in both equations should be opposites or equal. Let's align the coefficients:
Equation 1: 2x + y = 8
Equation 2: x - y = 1
To eliminate y, add the two equations directly:
(2x + y) + (x - y) = 8 + 1
Simplify:
2x + y + x - y = 9
Notice y and -y cancel out:
3x = 9
Divide both sides by 3:
x = 3
Substitute x = 3 into Equation 2 to find y:
3 - y = 1
Subtract 3 from both sides:
-y = -2
Multiply both sides by -1:
y = 2
The solution obtained via elimination is also (x, y) = (3, 2).
Comparison and Reflection
Both methods consistently lead to the same solution, illustrating their validity and reliability. The substitution method requires solving for one variable explicitly and substituting, which may be more straightforward when one equation is easily rearranged. Conversely, the elimination method can be more efficient when coefficients align conveniently, especially for larger systems or when dealing with multiple variables.
Personally, I prefer the elimination method in this case because it involves fewer steps and reduces the risk of algebraic mistakes during substitution. However, understanding both methods is essential, as each has situations where it is more effective.
Lessons Learned
Through completing this presentation, I learned the importance of choosing the appropriate method based on the specific system of equations. Practice with substitution and elimination enhances problem-solving versatility, providing multiple avenues to tackle systems efficiently. I also gained a deeper appreciation for algebraic manipulation and the significance of verifying solutions by substituting back into the original equations.
Conclusion
Mastering different methods for solving systems of equations is fundamental in mathematics and its applications. The substitution and elimination techniques, while different in approach, both lead to the same solution, reinforcing the correctness and robustness of these methods. Choosing the right method depends on the system's structure, and familiarity with both enhances overall problem-solving skills. Ultimately, understanding these methods equips students with essential tools to analyze relationships between variables systematically.
References
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- Larson, R., & Hostetler, R. (2018). Algebra 2. Cengage Learning.
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- Stewart, J. (2018). Calculus: Early Transcendentals. Cengage Learning.
- Strang, G. (2016). Introduction to Applied Mathematics. Wellesley-Cambridge Press.
- Kaczmarz, S. (1937). Approximate Solution of Systems of Linear Equations. International Journal of Control.
- Lay, D. C. (2021). Linear Algebra and Its Applications. Pearson.
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