Solving Y=-5x+30 And X=10 System Of Equations Step By Step
In mathematics, solving systems of equations is a fundamental skill. These systems involve two or more equations with multiple variables, and the goal is to find the values of the variables that satisfy all equations simultaneously. This article will provide a detailed walkthrough on how to solve a specific system of equations, offering a clear and comprehensive explanation suitable for students and anyone looking to refresh their algebra skills. The system we will tackle involves two equations: y = -5x + 30 and x = 10. Understanding how to approach such problems is crucial, as it forms the basis for more complex mathematical concepts and real-world applications.
Understanding Systems of Equations
Before diving into the solution, it's essential to grasp the concept of a system of equations. A system of equations is a set of two or more equations containing the same variables. The solution to the system is the set of values for the variables that make all the equations true. Graphically, the solution represents the point(s) where the lines or curves represented by the equations intersect. For linear equations, this intersection is a single point, indicating a unique solution. However, systems can also have no solutions (parallel lines) or infinitely many solutions (the same line).
There are several methods to solve systems of equations, including substitution, elimination, and graphing. Each method has its advantages, depending on the structure of the equations. The substitution method is particularly useful when one equation is already solved for one variable, as is the case in our system. This method involves substituting the expression for one variable from one equation into the other equation, thereby reducing the system to a single equation with one variable. Solving this equation provides the value of one variable, which can then be substituted back into either original equation to find the value of the other variable. This systematic approach ensures an accurate solution and is applicable to a wide range of systems of equations.
In the context of real-world applications, systems of equations can model various scenarios, such as determining the break-even point for a business, calculating the optimal mix of resources, or analyzing the trajectory of a projectile. The ability to solve these systems is therefore not only a mathematical skill but also a practical tool for problem-solving in various fields. By mastering the techniques for solving systems of equations, individuals can enhance their analytical abilities and approach complex problems with confidence.
Step-by-Step Solution
The given system of equations is:
- y = -5x + 30
- x = 10
This system is particularly straightforward to solve using the substitution method. Since the second equation already provides the value of x, we can substitute this value into the first equation to find the value of y. This simplifies the problem significantly, allowing us to quickly determine the solution.
Step 1: Substitute the Value of x
The second equation, x = 10, tells us that the value of x is 10. We substitute this value into the first equation:
- y = -5(10) + 30
This step is crucial as it replaces the variable x in the first equation with its numerical value, transforming the equation into one involving only the variable y. This transformation allows us to isolate y and solve for its value. The substitution method is effective because it systematically reduces the complexity of the system, making it easier to find the solution. By carefully substituting the known value, we ensure that the resulting equation is accurate and that the solution we obtain for y is correct.
Step 2: Simplify and Solve for y
Now, we simplify the equation:
- y = -50 + 30
- y = -20
This simplification involves performing the arithmetic operations to isolate y. First, we multiply -5 by 10, which gives -50. Then, we add 30 to -50. This straightforward calculation leads us to the value of y. The result, y = -20, is a critical part of the solution, as it provides the y-coordinate of the point where the two lines intersect on a graph. This step highlights the importance of careful arithmetic and attention to signs to ensure the accuracy of the final answer. By methodically simplifying the equation, we arrive at a clear and precise value for y, completing the process of solving for this variable.
The Solution
We have found that x = 10 and y = -20. Therefore, the solution to the system of equations is the ordered pair (10, -20). This ordered pair represents the point where the two lines described by the equations intersect on the Cartesian plane. The x-coordinate is 10, and the y-coordinate is -20. This solution satisfies both equations simultaneously, meaning that when we substitute these values into the original equations, both equations hold true. The solution is a unique point, indicating that the system has exactly one solution.
To verify the solution, we can substitute the values of x and y back into the original equations:
- y = -5x + 30 becomes -20 = -5(10) + 30, which simplifies to -20 = -50 + 30, and further to -20 = -20. This confirms that the first equation is satisfied.
- x = 10 remains 10 = 10, which is clearly true.
Since both equations are satisfied, the solution (10, -20) is correct. This verification step is an essential part of the problem-solving process, as it ensures that the solution is accurate and that no errors were made during the calculations. By checking the solution, we gain confidence in our answer and confirm our understanding of the system of equations. This thorough approach is crucial for success in mathematics and related fields.
Analyzing the Answer Choices
Now let's consider the multiple-choice options provided:
a. (-20, 10) b. (10, -20) c. (10, 4) d. (4, 10)
Based on our step-by-step solution, we found that the correct answer is (10, -20), which corresponds to option b. This highlights the importance of a systematic approach in solving mathematical problems. By carefully substituting and simplifying, we arrived at the correct solution and were able to confidently identify the correct answer among the given choices.
Option a, (-20, 10), is incorrect because it reverses the x and y values and uses the wrong sign for y. Substituting these values into the equations would not satisfy both equations simultaneously. This option might be a result of a misunderstanding of the substitution process or a mistake in arithmetic.
Option c, (10, 4), is also incorrect. While the x-value is correct, the y-value is not. If we substitute x = 10 into the first equation, we get y = -5(10) + 30 = -20, not 4. This option could arise from an error in the simplification step or a miscalculation of the arithmetic.
Option d, (4, 10), is incorrect because both the x and y values are wrong. The x-value should be 10 according to the second equation, and the y-value should be -20 as we calculated. This option might indicate a more significant misunderstanding of the problem or a failure to correctly apply the substitution method.
By analyzing the incorrect options, we can see common mistakes that students might make when solving systems of equations. Understanding these potential errors can help students avoid them in the future and reinforce the correct methods for solving such problems. The process of elimination, combined with a clear and systematic solution, is a powerful tool for ensuring accuracy in mathematical problem-solving.
Conclusion
In conclusion, the solution to the system of equations y = -5x + 30 and x = 10 is (10, -20). We arrived at this solution by using the substitution method, which involved substituting the value of x from the second equation into the first equation and then solving for y. This step-by-step process is a reliable way to solve systems of equations, especially when one equation is already solved for a variable.
The ability to solve systems of equations is a crucial skill in mathematics and has wide-ranging applications in various fields. From determining the equilibrium point in economics to calculating trajectories in physics, systems of equations provide a powerful tool for modeling and solving real-world problems. Mastering the techniques for solving these systems not only enhances mathematical proficiency but also improves problem-solving abilities in general.
The key to success in solving systems of equations lies in a clear understanding of the methods available and a systematic approach to the problem. By carefully substituting, simplifying, and verifying the solution, students can confidently tackle even complex systems of equations. This article has provided a detailed example of how to solve a specific system, offering a model for approaching similar problems in the future. Through practice and a solid understanding of the underlying concepts, anyone can develop the skills necessary to solve systems of equations effectively.
