Solving Systems Of Equations 3x + 2y = 5 And 6x + 4y = 0
In mathematics, solving a system of equations is a fundamental skill with applications across various fields, including physics, engineering, economics, and computer science. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. There are several methods for solving systems of equations, including substitution, elimination, and graphical methods. Each method has its advantages and disadvantages, and the choice of method often depends on the specific system of equations being solved. Understanding how to solve systems of equations is crucial for anyone pursuing studies or careers in STEM fields, as well as for solving practical problems in everyday life.
In this article, we will explore a particular system of equations and demonstrate how to solve it using a convenient method. We will also discuss the importance of checking solutions and interpreting the results in the context of the problem. This comprehensive guide aims to provide a clear and thorough understanding of the process involved in solving systems of equations, ensuring that readers can confidently tackle similar problems in the future. By mastering these techniques, individuals can enhance their problem-solving abilities and apply these skills to a wide range of real-world scenarios. Let's dive into the specifics of solving systems of equations and uncover the strategies that make this mathematical process both accessible and effective.
To solve a system of equations, you need to find the values of the variables that satisfy all equations in the system simultaneously. Letβs consider the given system of equations:
This system consists of two linear equations with two variables, x and y. Linear equations are equations in which the highest power of the variables is 1. The graph of a linear equation is a straight line. To solve this system, we need to find the values of x and y that make both equations true. We can use several methods to solve systems of equations, such as substitution, elimination, or graphing. In this case, the elimination method is particularly convenient.
Before we proceed, it's essential to understand the nature of the system. A system of equations can have one unique solution, infinitely many solutions, or no solution. The number of solutions depends on the relationship between the equations. If the equations represent intersecting lines, there is one unique solution. If the equations represent the same line, there are infinitely many solutions. If the equations represent parallel lines, there is no solution. Recognizing these possibilities is crucial in problem-solving, as it guides the approach and interpretation of results. Let's delve into the elimination method to determine the solution, if any, for our specific system. This method will help us systematically eliminate one variable, making it easier to solve for the other.
When solving systems of equations, selecting the appropriate method is crucial for efficiency and accuracy. The elimination method is particularly well-suited for systems where the coefficients of one of the variables are multiples of each other or can easily be made multiples through multiplication. In our system:
we observe that the coefficients of x in the two equations are 3 and 6, respectively. Similarly, the coefficients of y are 2 and 4. This relationship makes the elimination method an ideal choice. The goal of the elimination method is to eliminate one variable by adding or subtracting the equations. To do this, we need to make the coefficients of one variable the same (or additive inverses) in both equations.
The elimination method is often preferred when the equations are in standard form (Ax + By = C). It provides a systematic approach to solving the system by reducing it to a single equation with one variable. This method is especially effective when dealing with linear equations, as it simplifies the process and minimizes the chances of making algebraic errors. The choice of the elimination method also allows us to quickly identify if the system has a unique solution, no solution, or infinitely many solutions, based on the outcome of the elimination process. Now, letβs apply the elimination method to our system and see how it works step-by-step.
To apply the elimination method, we aim to make the coefficients of either x or y the same (or additive inverses) in both equations. Looking at our system:
we can easily make the coefficients of x the same by multiplying the first equation by -2. This will give us -6x in the first equation, which is the additive inverse of 6x in the second equation. Multiplying the first equation by -2, we get:
Now, our system looks like this:
Next, we add the two equations together:
The x and y terms cancel out, leaving us with:
This is a contradiction, as 0 cannot equal -10. This result indicates that the system of equations has no solution. A system with no solution means that the lines represented by the equations are parallel and do not intersect. This step-by-step application of the elimination method has clearly shown that there is no set of values for x and y that can satisfy both equations simultaneously.
The result 0 = -10 obtained from the elimination method is a crucial indicator that the system of equations has no solution. This contradiction arises because the two equations represent parallel lines. Parallel lines have the same slope but different y-intercepts, meaning they will never intersect. Therefore, there is no point (x, y) that lies on both lines simultaneously.
To further understand this, letβs look at the original equations:
If we try to solve the first equation for y, we get:
y = - rac{3}{2}x + rac{5}{2}
Similarly, for the second equation:
y = - rac{6}{4}x
y = - rac{3}{2}x
We can see that both lines have the same slope (-3/2) but different y-intercepts (5/2 and 0). This confirms that the lines are parallel and do not intersect, hence no solution exists for the system of equations.
In conclusion, when applying the elimination method leads to a contradiction like 0 = -10, it signifies that the system of equations is inconsistent and has no solution. This understanding is vital for accurately interpreting the results of mathematical problem-solving and applying these concepts to real-world scenarios where similar situations may arise.
The system of equations has no solution. This conclusion is reached through the elimination method, which resulted in a contradiction (0 = -10). This indicates that the two equations represent parallel lines, which do not intersect. Therefore, there is no pair of (x, y) values that can satisfy both equations simultaneously. The correct choice is:
A. There is no solution.
