What Is The Solution \[$(q, R)\$\] To This System Of Linear Equations?$\[ \begin{array}{l} 12q + 3r = 15 \\ -4q - 4r = -44 \end{array} \\]A. \[$(-18, 29)\$\]B. \[$(-2, 13)\$\]C. \[$(8, -1)\$\]D. \[$(15,
Introduction
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 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 to the system.
The System of Linear Equations
The system of linear equations we will be solving is given by:
12q + 3r = 15 -4q - 4r = -44
Step 1: Multiply the Two Equations by Necessary Multiples
To eliminate one of the variables, we need to make the coefficients of either q or r the same in both equations but with opposite signs. We can do this by multiplying the two equations by necessary multiples.
Let's multiply the first equation by 4 and the second equation by 3:
48q + 12r = 60 -12q - 12r = -132
Step 2: Add the Two Equations
Now that we have the coefficients of q and r as 48 and -12 respectively, we can add the two equations to eliminate q.
(48q + 12r) + (-12q - 12r) = 60 + (-132) 36q = -72
Step 3: Solve for q
Now that we have the equation 36q = -72, we can solve for q by dividing both sides by 36.
q = -72/36 q = -2
Step 4: Substitute the Value of q into One of the Original Equations
Now that we have the value of q, we can substitute it into one of the original equations to solve for r. Let's substitute q = -2 into the first equation:
12(-2) + 3r = 15 -24 + 3r = 15
Step 5: Solve for r
Now that we have the equation -24 + 3r = 15, we can solve for r by adding 24 to both sides and then dividing both sides by 3.
3r = 39 r = 39/3 r = 13
Step 6: Write the Solution as an Ordered Pair
Now that we have the values of q and r, we can write the solution as an ordered pair (q, r).
(q, r) = (-2, 13)
Conclusion
In this article, we solved a system of two linear equations with two variables using the method of substitution and elimination. We multiplied the two equations by necessary multiples, added the two equations to eliminate one of the variables, solved for the other variable, and then wrote the solution as an ordered pair. The solution to the system of linear equations is (q, r) = (-2, 13).
Discussion
The solution to the system of linear equations is (q, r) = (-2, 13). This means that when q = -2 and r = 13, both equations are satisfied. The solution can be verified by plugging the values of q and r into both equations.
Final Answer
The final answer is .
Introduction
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 answer some frequently asked questions (FAQs) about solving systems of linear equations.
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 that it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: What are the methods for solving systems of linear equations?
A: There are several methods for solving systems of linear equations, including:
- Substitution method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
- Elimination method: This method involves adding or subtracting the two equations to eliminate one of the variables.
- Graphical method: This method involves graphing the two equations on a coordinate plane and finding the point of intersection.
- Matrix method: This method involves using matrices to represent the system of equations and then solving for the variables.
Q: What is the difference between a system of linear equations and a system of nonlinear equations?
A: A system of linear equations is a set 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, is a set of nonlinear equations, which means that each equation cannot be written in the form ax + by = c.
Q: How do I know which method to use to solve a system of linear equations?
A: The choice of method depends on the specific system of equations and the variables involved. If the system has two variables and two equations, the substitution and elimination methods are often the most efficient. If the system has more than two variables or more than two equations, the matrix method may be more suitable.
Q: Can a system of linear equations have no solution?
A: Yes, a system of linear equations can have no solution. This occurs when the two equations are inconsistent, meaning that they cannot both be true at the same time.
Q: Can a system of linear equations have infinitely many solutions?
A: Yes, a system of linear equations can have infinitely many solutions. This occurs when the two equations are dependent, meaning that one equation is a multiple of the other.
Q: How do I check my work when solving a system of linear equations?
A: To check your work, plug the values of the variables back into both equations and make sure that both equations are satisfied.
Q: What are some common mistakes to avoid when solving systems of linear equations?
A: Some common mistakes to avoid include:
- Not checking the work: Make sure to plug the values of the variables back into both equations to check that both equations are satisfied.
- Not using the correct method: Choose the method that is most suitable for the specific system of equations.
- Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when solving the equations.
Conclusion
In this article, we have answered some frequently asked questions (FAQs) about solving systems of linear equations. We have discussed the different methods for solving systems of linear equations, the difference between a system of linear equations and a system of nonlinear equations, and some common mistakes to avoid. By following these tips and techniques, you can become more confident and proficient in solving systems of linear equations.
Final Answer
The final answer is .