Find The Value Of $Q$ In The Following System So That The Solution To The System Is $(4,2)$.$\[ \begin{array}{l} 3x - 2y = 8 \\ 2x + 3y = Q \end{array} \\]A. 14 B. 8 C. 12 D. 10
Introduction
In this article, we will explore how to find the value of Q in a linear system of equations, given that the solution to the system is (4,2). We will use the method of substitution to solve for Q.
The System of Equations
The system of equations is given as:
{ \begin{array}{l} 3x - 2y = 8 \\ 2x + 3y = Q \end{array} \}
Substitution Method
To solve for Q, we can use the substitution method. We will substitute the value of x and y from the solution (4,2) into the second equation.
Substituting x and y
We know that x = 4 and y = 2. We can substitute these values into the second equation:
2(4) + 3(2) = Q
Simplifying the Equation
Now, we can simplify the equation:
8 + 6 = Q
Solving for Q
Now, we can solve for Q:
Q = 14
Conclusion
Therefore, the value of Q in the system of equations is 14.
Why is this the Correct Answer?
To verify that this is the correct answer, we can substitute the value of Q back into the system of equations and check if the solution (4,2) satisfies both equations.
Verifying the Solution
We can substitute Q = 14 into the second equation:
2x + 3y = 14
Substituting x = 4 and y = 2, we get:
2(4) + 3(2) = 14
8 + 6 = 14
This is true, so the solution (4,2) satisfies both equations.
Conclusion
Therefore, the value of Q in the system of equations is indeed 14.
What is the Importance of Solving Linear Systems of Equations?
Solving linear systems of equations is an important skill in mathematics and has many real-world applications. It is used in a variety of fields, including physics, engineering, economics, and computer science.
Real-World Applications
Some real-world applications of solving linear systems of equations include:
- Physics: Solving linear systems of equations is used to describe the motion of objects in physics.
- Engineering: Solving linear systems of equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: Solving linear systems of equations is used to model economic systems and make predictions about economic outcomes.
- Computer Science: Solving linear systems of equations is used in computer graphics, game development, and machine learning.
Conclusion
In conclusion, solving linear systems of equations is an important skill in mathematics that has many real-world applications. By using the substitution method, we can find the value of Q in a system of equations, given that the solution to the system is (4,2). The value of Q is 14.
Final Answer
The final answer is .
Introduction
In our previous article, we explored how to find the value of Q in a linear system of equations, given that the solution to the system is (4,2). We used the method of substitution to solve for Q. In this article, we will answer some frequently asked questions about solving linear systems of equations.
Q: What is a linear system of equations?
A: A linear system of equations is a set of two or more linear equations that are solved simultaneously. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: What are the different methods for solving linear systems of equations?
A: There are several methods for solving linear systems of equations, including:
- Substitution Method: This method involves substituting the value of one variable from one equation into the other equation.
- Elimination Method: This method involves adding or subtracting the equations to eliminate one of the variables.
- Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
Q: How do I choose which method to use?
A: The choice of method depends on the type of system and the variables involved. If the system has two equations with two variables, the substitution or elimination method may be used. If the system has more than two equations or variables, the graphical method may be used.
Q: What is the importance of solving linear systems of equations?
A: Solving linear systems of equations is an important skill in mathematics and has many real-world applications. It is used in a variety of fields, including physics, engineering, economics, and computer science.
Q: Can you give an example of a real-world application of solving linear systems of equations?
A: Yes, one example is in physics, where solving linear systems of equations is used to describe the motion of objects. For example, if you have a car moving at a certain speed and direction, you can use linear systems of equations to predict its position and velocity at a future time.
Q: How do I verify that the solution to a linear system of equations is correct?
A: To verify that the solution to a linear system of equations is correct, you can substitute the values of the variables back into the original equations and check if the equations are satisfied.
Q: What are some common mistakes to avoid when solving linear systems of equations?
A: Some common mistakes to avoid when solving linear systems of equations include:
- Not checking the validity of the solution: Make sure to verify that the solution satisfies both equations.
- Not using the correct method: Choose the method that is best suited for the type of system and variables involved.
- Not simplifying the equations: Simplify the equations as much as possible to make it easier to solve.
Q: Can you give some tips for solving linear systems of equations?
A: Yes, here are some tips:
- Read the problem carefully: Make sure to understand what is being asked and what the variables represent.
- Choose the correct method: Choose the method that is best suited for the type of system and variables involved.
- Simplify the equations: Simplify the equations as much as possible to make it easier to solve.
- Check the validity of the solution: Verify that the solution satisfies both equations.
Conclusion
In conclusion, solving linear systems of equations is an important skill in mathematics that has many real-world applications. By understanding the different methods for solving linear systems of equations and following some simple tips, you can become proficient in solving these types of problems.
Final Answer
The final answer is .