Solving Systems Of Equations Step-by-Step Guide
In the realm of mathematics, solving systems of equations is a fundamental skill with far-reaching applications. From determining the intersection of lines on a graph to modeling complex relationships in science and engineering, the ability to solve systems of equations is indispensable. This article delves into the intricacies of solving a specific system of equations, providing a step-by-step approach and offering insights into the underlying concepts. We will explore the given system, discuss the methods for solving it, and arrive at the correct solution while ensuring a clear understanding of the process.
Understanding the System of Equations
At the heart of our discussion lies the following system of equations:
y = 2x - 3.5
x - 2y = -14
This system comprises two linear equations, each representing a straight line on a Cartesian plane. The solution to this system is the point (or points) where these lines intersect. In other words, it's the set of values for x and y that satisfy both equations simultaneously. Linear equations are foundational in mathematics, forming the basis for more advanced concepts. A linear equation, in its simplest form, is an equation that can be written as ax + by = c, where a, b, and c are constants, and x and y are variables. The graph of a linear equation is a straight line. When we have a system of two linear equations, we are essentially looking for the point where these two lines intersect. This intersection point, if it exists, represents the solution to the system. There are three possible scenarios when dealing with a system of two linear equations: the lines may intersect at one point (resulting in a unique solution), they may be parallel and never intersect (resulting in no solution), or they may be the same line (resulting in infinitely many solutions). Understanding the nature of these equations is the first step towards finding their solution. Before diving into the methods for solving this system, let’s briefly discuss the significance of understanding the equations themselves. The first equation, y = 2x - 3.5, is in slope-intercept form, which immediately tells us that the slope of the line is 2 and the y-intercept is -3.5. The second equation, x - 2y = -14, can be rearranged into slope-intercept form as well, which will help us visualize the relationship between the two lines. This preliminary analysis is crucial because it can often provide insights into the nature of the solution, such as whether the lines are likely to intersect at a single point, are parallel, or are coincident.
Methods for Solving Systems of Equations
Several methods are available for solving systems of equations, each with its own advantages and applications. The two most common methods are:
- Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation. This eliminates one variable, allowing us to solve for the remaining variable.
- Elimination: This method involves manipulating the equations to make the coefficients of one variable opposites, then adding the equations together. This eliminates one variable, allowing us to solve for the remaining variable.
For the given system of equations, the substitution method appears to be the most straightforward approach, given that the first equation is already solved for y. This makes the substitution process less cumbersome and reduces the chances of making algebraic errors. However, it's important to note that the choice of method often depends on the specific form of the equations in the system. In some cases, the elimination method might be more efficient, especially if the coefficients of one of the variables are easily made opposites through multiplication. Both methods are based on the fundamental principle of maintaining equality. When we perform an operation on one side of an equation, we must perform the same operation on the other side to preserve the balance. This principle ensures that the solutions we obtain are valid and consistent with the original equations. In addition to substitution and elimination, graphical methods can also be used to solve systems of equations. By plotting the equations on a graph, the intersection point can be visually identified, providing a geometric interpretation of the solution. While graphical methods may not always yield precise solutions, they can be valuable for gaining a qualitative understanding of the system and for verifying solutions obtained through algebraic methods. Moreover, technology such as graphing calculators and computer software can greatly assist in solving systems of equations, especially for more complex systems involving multiple variables and equations. These tools can perform the algebraic manipulations and graphical representations, allowing students and professionals to focus on the interpretation and application of the solutions. Ultimately, the selection of the most appropriate method for solving a system of equations depends on the specific characteristics of the system, the desired level of precision, and the available tools and resources.
Applying the Substitution Method
Let's apply the substitution method to our system of equations.
We have:
y = 2x - 3.5 (Equation 1)
x - 2y = -14 (Equation 2)
Since Equation 1 is already solved for y, we can substitute the expression 2x - 3.5 for y in Equation 2:
x - 2(2x - 3.5) = -14
Now, we solve for x:
x - 4x + 7 = -14
-3x + 7 = -14
-3x = -21
x = 7
With x = 7, we can substitute this value back into Equation 1 to find y:
y = 2(7) - 3.5
y = 14 - 3.5
y = 10.5
Therefore, the solution to the system of equations is (7, 10.5). The substitution method is a powerful technique for solving systems of equations, particularly when one equation is already solved for one variable or can be easily rearranged to isolate a variable. The key to this method is to carefully substitute the expression for the isolated variable into the other equation, ensuring that all instances of the variable are replaced. This process effectively reduces the system to a single equation with one variable, which can then be solved using standard algebraic techniques. Once the value of one variable is found, it can be substituted back into either of the original equations to determine the value of the other variable. It's crucial to pay close attention to the order of operations and to distribute any coefficients correctly to avoid errors. A common mistake is to forget to distribute a negative sign when substituting an expression into an equation. For example, in our case, when we substituted 2x - 3.5 for y in the second equation, we had to distribute the -2 across both terms inside the parentheses. This step is critical to obtaining the correct solution. After solving for both variables, it's always a good practice to check the solution by substituting the values back into both original equations to ensure that they are satisfied. This step helps to verify the accuracy of the solution and to catch any potential errors made during the substitution or simplification process. The substitution method is not only useful for solving systems of linear equations but can also be applied to systems involving non-linear equations, such as quadratic or exponential equations, provided that one of the equations can be solved for one variable in terms of the other. In such cases, the substitution process may lead to more complex equations, but the underlying principle remains the same: to reduce the system to a single equation with one variable.
Analyzing the Answer Choices
Now, let's examine the provided answer choices:
A. (-7, 3.5) B. (3.5, -7) C. (7, 10.5) D. (10.5, 7)
Our solution (7, 10.5) corresponds to answer choice C. Each answer choice represents a potential solution to the system of equations, and the task is to identify the pair of values that satisfy both equations simultaneously. A straightforward way to verify the correctness of an answer choice is to substitute the x and y values into both equations and check if the equations hold true. If the values satisfy both equations, then the answer choice is a valid solution. If they do not, then the answer choice is incorrect. This verification process is especially useful when dealing with multiple-choice questions, as it allows you to eliminate incorrect options and narrow down the possibilities. For instance, let's consider answer choice A (-7, 3.5). Substituting these values into the first equation, y = 2x - 3.5, we get 3.5 = 2(-7) - 3.5, which simplifies to 3.5 = -14 - 3.5, which is clearly false. Therefore, we can eliminate answer choice A. Similarly, we can test the other answer choices until we find the one that satisfies both equations. In the case of answer choice C (7, 10.5), substituting these values into the first equation, we get 10.5 = 2(7) - 3.5, which simplifies to 10.5 = 14 - 3.5, which is true. Substituting these values into the second equation, x - 2y = -14, we get 7 - 2(10.5) = -14, which simplifies to 7 - 21 = -14, which is also true. Therefore, answer choice C is the correct solution. Analyzing the answer choices in this way not only helps in confirming the correctness of the solution but also reinforces the understanding of what it means for a pair of values to be a solution to a system of equations. It emphasizes the concept that the solution must simultaneously satisfy all equations in the system. Furthermore, this approach can be particularly beneficial in test-taking situations where time is limited, as it provides a systematic way to evaluate the options and arrive at the correct answer.
Conclusion
The solution to the system of equations
y = 2x - 3.5
x - 2y = -14
is (7, 10.5), which corresponds to answer choice C. This article has demonstrated the application of the substitution method for solving systems of linear equations. Understanding the underlying principles and practicing different methods are crucial for mastering this fundamental mathematical skill. Solving systems of equations is a cornerstone of mathematical problem-solving, with applications spanning various fields, from algebra and calculus to physics and economics. The ability to solve systems of equations empowers individuals to model real-world scenarios, analyze relationships between variables, and make informed decisions. Whether it's determining the equilibrium point in a supply-demand model or calculating the trajectory of a projectile, the techniques for solving systems of equations provide a powerful toolset for tackling complex problems. The process of solving systems of equations involves a combination of algebraic manipulation, logical reasoning, and attention to detail. There are several methods available, each with its own strengths and weaknesses, and the choice of method often depends on the specific characteristics of the system. The substitution method, as demonstrated in this article, is particularly effective when one of the equations is already solved for one variable or can be easily rearranged to isolate a variable. The elimination method, on the other hand, is well-suited for systems where the coefficients of one of the variables are easily made opposites through multiplication. Graphical methods can also provide valuable insights into the nature of the solutions, allowing for a visual representation of the intersection points of the equations. Regardless of the method employed, the key is to maintain accuracy and consistency in the algebraic manipulations. A small error in a single step can propagate through the entire solution process, leading to an incorrect answer. Therefore, it's essential to double-check each step and to verify the final solution by substituting the values back into the original equations. Moreover, the ability to interpret the solutions in the context of the problem is crucial. In many real-world applications, the solutions represent physical quantities or economic indicators, and understanding their meaning is just as important as finding the numerical values. For instance, in a system of equations representing the motion of two objects, the solution might represent the time and position at which the objects collide. In an economic model, the solution might represent the prices and quantities at which markets clear. Therefore, solving systems of equations is not just a mathematical exercise but a skill that can be applied to a wide range of practical problems.
