Which Statement Describes The Solution To The System Of Equations?$\[ \begin{aligned} 2h + 8k &= 6 \\ -5h - 20k &= -15 \end{aligned} \\]A. The System Has Exactly One Solution. B. The System Has Exactly Two Solutions. C. The System Has No
**Which Statement Describes the Solution to the System of Equations?**
Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. In this article, we will explore the solution to a system of two linear equations with two variables. We will analyze the given system of equations and determine which statement describes the solution.
The given system of equations is:
{ \begin{aligned} 2h + 8k &= 6 \\ -5h - 20k &= -15 \end{aligned} \}
Step 1: Analyze the System of Equations
To determine the solution to the system of equations, we need to analyze the given equations. We can start by examining the coefficients of the variables h and k. The first equation has a coefficient of 2 for h and 8 for k, while the second equation has a coefficient of -5 for h and -20 for k.
Step 2: Determine the Type of System
We can determine the type of system by examining the coefficients of the variables. If the coefficients are proportional, then the system is consistent and has a unique solution. If the coefficients are not proportional, then the system is inconsistent and has no solution.
Step 3: Solve the System of Equations
To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.
Elimination Method
We can multiply the first equation by 5 and the second equation by 2 to make the coefficients of h in both equations equal.
{ \begin{aligned} 10h + 40k &= 30 \\ -10h - 40k &= -30 \end{aligned} \}
Now, we can add both equations to eliminate the variable h.
{ \begin{aligned} (10h + 40k) + (-10h - 40k) &= 30 + (-30) \\ 0 &= 0 \end{aligned} \}
As we can see, the resulting equation is a contradiction, which means that the system of equations has no solution.
Based on the analysis, we can conclude that the system of equations has no solution. Therefore, the correct statement is:
C. The system has no solution
Q: What is the solution to the system of equations? A: The system of equations has no solution.
Q: Why is the system of equations inconsistent? A: The system of equations is inconsistent because the coefficients of the variables are not proportional.
Q: What is the method used to solve the system of equations? A: The elimination method is used to solve the system of equations.
Q: What is the result of adding both equations in the elimination method? A: The result of adding both equations is a contradiction, which means that the system of equations has no solution.
Q: What is the correct statement that describes the solution to the system of equations? A: The correct statement is C. The system has no solution.
Q: What is the difference between a consistent and inconsistent system of equations? A: A consistent system of equations has a unique solution, while an inconsistent system of equations has no solution.
Q: How do you determine the type of system of equations? A: You can determine the type of system by examining the coefficients of the variables. If the coefficients are proportional, then the system is consistent. If the coefficients are not proportional, then the system is inconsistent.
Q: What is the elimination method? A: The elimination method is a method used to solve systems of linear equations by eliminating one of the variables.
Q: What is the result of adding both equations in the elimination method? A: The result of adding both equations is a contradiction, which means that the system of equations has no solution.
In conclusion, the system of equations has no solution. The elimination method was used to solve the system of equations, and the result was a contradiction, which means that the system of equations has no solution. The correct statement that describes the solution to the system of equations is C. The system has no solution.