Solve The System Of Equations.${ \begin{cases} 2x + 8y = 22 \ 3x + 12y = 33 \end{cases} }$Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. There Is One Solution. The Solution Of The
Introduction
Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the solution.
What are Systems of Linear Equations?
A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is a linear equation, which means it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are variables.
The System of Equations
The system of equations we will be solving is:
{ \begin{cases} 2x + 8y = 22 \\ 3x + 12y = 33 \end{cases} \}
Step 1: Write Down the Equations
The first step in solving a system of linear equations is to write down the equations. In this case, we have two equations:
- 2x + 8y = 22
- 3x + 12y = 33
Step 2: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to multiply the equations by necessary multiples. We can multiply the first equation by 3 and the second equation by 2.
- 6x + 24y = 66
- 6x + 24y = 66
Step 3: Subtract the Second Equation from the First Equation
Now that we have the same coefficients for x in both equations, we can subtract the second equation from the first equation to eliminate x.
(6x + 24y) - (6x + 24y) = 66 - 66
This simplifies to:
0 = 0
Step 4: Conclusion
Since the equation 0 = 0 is true for all values of x and y, we can conclude that the system of equations has infinitely many solutions. This means that there are an infinite number of points that satisfy both equations.
Why is this the case?
The reason why the system of equations has infinitely many solutions is that the two equations are actually the same equation. When we multiplied the first equation by 3 and the second equation by 2, we ended up with the same equation:
6x + 24y = 66
This means that the two equations are linearly dependent, and there are an infinite number of solutions.
Conclusion
In conclusion, solving systems of linear equations is a crucial skill for students and professionals alike. By using the method of substitution and elimination, we can find the solution to a system of two linear equations with two variables. In this article, we solved a system of equations that had infinitely many solutions. We hope that this article has provided you with a better understanding of how to solve systems of linear equations.
Final Answer
The final answer is:
There is no unique solution to the system of equations. The system has infinitely many solutions.
Discussion
This problem is a great example of how to use the method of substitution and elimination to solve systems of linear equations. It also highlights the importance of checking for linear dependence between the equations. If the equations are linearly dependent, then the system has infinitely many solutions.
Related Topics
- Solving systems of linear equations using substitution
- Solving systems of linear equations using elimination
- Linear dependence and independence
- Infinite solutions to systems of linear equations
Practice Problems
- Solve the system of equations:
{ \begin{cases} x + 2y = 6 \\ 2x + 4y = 12 \end{cases} \}
- Solve the system of equations:
{ \begin{cases} 3x + 2y = 12 \\ 6x + 4y = 24 \end{cases} \}
- Solve the system of equations:
{ \begin{cases} x + y = 4 \\ 2x + 2y = 8 \end{cases} \}
Solutions
- The solution to the system of equations is x = 2 and y = 2.
- The solution to the system of equations is x = 2 and y = 2.
- The solution to the system of equations is x = 2 and y = 2.
Conclusion
Q: What is a system of linear equations?
A: A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is a linear equation, which means it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are variables.
Q: How do I solve a system of linear equations?
A: There are several methods to solve a system of linear equations, including:
- Substitution method: This involves solving one equation for one variable and then substituting that expression into the other equation.
- Elimination method: This involves adding or subtracting the equations to eliminate one of the variables.
- Graphical method: This involves graphing the equations on a coordinate plane and finding the point of intersection.
Q: What is the difference between a system of linear equations and a system of nonlinear equations?
A: A system of linear equations consists of linear equations, which means that each equation can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are variables. A system of nonlinear equations, on the other hand, consists of nonlinear equations, which means that each equation cannot be written in the form:
ax + by = c
Q: How do I determine if a system of linear equations has a unique solution, infinitely many solutions, or no solution?
A: To determine if a system of linear equations has a unique solution, infinitely many solutions, or no solution, you can use the following criteria:
- If the two equations are parallel, then the system has no solution.
- If the two equations are identical, then the system has infinitely many solutions.
- If the two equations are not parallel and not identical, then the system has a unique solution.
Q: What is the importance of solving systems of linear equations?
A: Solving systems of linear equations is an important skill in mathematics and has many real-world applications, including:
- Physics: Solving systems of linear equations is used to model real-world problems, such as the motion of objects under the influence of gravity.
- Engineering: Solving systems of linear equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: Solving systems of linear equations is used to model economic systems and make predictions about future economic trends.
Q: How do I choose the best method for solving a system of linear equations?
A: The best method for solving a system of linear equations depends on the specific problem and the variables involved. Some common methods include:
- Substitution method: This is a good method to use when one of the variables is easily isolated.
- Elimination method: This is a good method to use when the coefficients of one of the variables are the same in both equations.
- Graphical method: This is a good method to use when the equations are easy to graph and the point of intersection is clear.
Q: What are some common mistakes to avoid when solving systems of linear equations?
A: Some common mistakes to avoid when solving systems of linear equations include:
- Not checking for linear dependence between the equations.
- Not checking for parallel or identical equations.
- Not using the correct method for solving the system.
- Not checking the solution for consistency with the original equations.
Q: How do I check my solution for consistency with the original equations?
A: To check your solution for consistency with the original equations, you can plug the values of the variables back into the original equations and check if they are true. If they are true, then the solution is consistent with the original equations. If they are not true, then the solution is not consistent with the original equations.