Solve The Following System Of Equations:$\[ \begin{array}{l} -4x + 3y = 7 \\ 2x - 5y = 7 \end{array} \\]A. (4, -3) B. (-4, 3) C. (4, 3) D. (-4, -3)
Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. 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.
The System of Equations
The given system of equations is:
{ \begin{array}{l} -4x + 3y = 7 \\ 2x - 5y = 7 \end{array} \}
Step 1: Multiply the Two Equations by Necessary Multiples
To eliminate one of the variables, we need to multiply the two equations by necessary multiples such that the coefficients of either x or y will be the same in both equations. Let's multiply the first equation by 5 and the second equation by 3.
{ \begin{array}{l} -20x + 15y = 35 \\ 6x - 15y = 21 \end{array} \}
Step 2: Add the Two Equations
Now, we can add the two equations to eliminate the variable y.
{ \begin{array}{l} -20x + 15y + 6x - 15y = 35 + 21 \\ -14x = 56 \end{array} \}
Step 3: Solve for x
Now, we can solve for x by dividing both sides of the equation by -14.
{ \begin{array}{l} x = -56/14 \\ x = -4 \end{array} \}
Step 4: Substitute the Value of x into One of the Original Equations
Now that we have the value of x, we can substitute it into one of the original equations to solve for y. Let's substitute x = -4 into the first equation.
{ \begin{array}{l} -4x + 3y = 7 \\ -4(-4) + 3y = 7 \\ 16 + 3y = 7 \end{array} \}
Step 5: Solve for y
Now, we can solve for y by subtracting 16 from both sides of the equation and then dividing both sides by 3.
{ \begin{array}{l} 3y = -9 \\ y = -9/3 \\ y = -3 \end{array} \}
Conclusion
Therefore, the solution to the system of equations is x = -4 and y = -3.
Answer
The correct answer is D. (-4, -3).
Discussion
This problem is a classic example of a system of linear equations with two variables. The method of substitution and elimination is a powerful tool for solving such systems. By multiplying the two equations by necessary multiples and then adding them, we were able to eliminate one of the variables and solve for the other. This problem requires a good understanding of algebraic manipulations and the ability to apply the method of substitution and elimination.
Real-World Applications
Systems of linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, systems of linear equations can be used to model the motion of objects under the influence of forces. In engineering, systems of linear equations can be used to design and optimize systems such as electrical circuits and mechanical systems. In economics, systems of linear equations can be used to model the behavior of economic systems and make predictions about future trends.
Tips and Tricks
When solving systems of linear equations, it is essential to follow the order of operations and to use the method of substitution and elimination correctly. It is also important to check the solution by substituting the values of x and y back into the original equations. Additionally, it is helpful to use graphing tools or software to visualize the solution and to check for any errors.
Common Mistakes
One common mistake when solving systems of linear equations is to forget to multiply the equations by necessary multiples or to add the equations incorrectly. Another common mistake is to substitute the values of x and y back into the original equations incorrectly. To avoid these mistakes, it is essential to follow the order of operations and to use the method of substitution and elimination correctly.
Conclusion
Introduction
In our previous article, we discussed how to solve a system of linear equations using the method of substitution and elimination. In this article, we will answer some frequently asked questions 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 are solved simultaneously to find the values of the variables.
Q: What are the different methods for solving systems of linear equations?
A: There are two main methods for solving systems of linear equations: the method of substitution and the method of elimination. The method of substitution involves substituting the value of one variable into the other equation, while the method of elimination involves adding or subtracting the two equations to eliminate one of the variables.
Q: What is the difference between the method of substitution and the method of elimination?
A: The main difference between the method of substitution and the method of elimination is the way in which the equations are manipulated. In the method of substitution, the value of one variable is substituted into the other equation, while in the method of elimination, the two equations are added or subtracted to eliminate one of the variables.
Q: How do I know which method to use?
A: The choice of method depends on the specific problem and the values of the coefficients. If the coefficients are simple and easy to work with, the method of substitution may be the best choice. However, if the coefficients are complex or difficult to work with, the method of elimination may be the better option.
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:
- Forgetting to multiply the equations by necessary multiples
- Adding the equations incorrectly
- Substituting the values of x and y back into the original equations incorrectly
- Not checking the solution by substituting the values of x and y back into the original equations
Q: How do I check my solution?
A: To check your solution, substitute the values of x and y back into the original equations and make sure that the equations are true. If the equations are true, then your solution is correct.
Q: What are some real-world applications of systems of linear equations?
A: Systems of linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, systems of linear equations can be used to model the motion of objects under the influence of forces. In engineering, systems of linear equations can be used to design and optimize systems such as electrical circuits and mechanical systems. In economics, systems of linear equations can be used to model the behavior of economic systems and make predictions about future trends.
Q: How do I use graphing tools or software to visualize the solution?
A: To visualize the solution, use graphing tools or software to plot the two equations on a coordinate plane. The point of intersection of the two lines represents the solution to the system of equations.
Q: What are some tips for solving systems of linear equations?
A: Some tips for solving systems of linear equations include:
- Follow the order of operations
- Use the method of substitution and elimination correctly
- Check the solution by substituting the values of x and y back into the original equations
- Use graphing tools or software to visualize the solution
Conclusion
In conclusion, solving a system of linear equations requires a good understanding of algebraic manipulations and the ability to apply the method of substitution and elimination. By following the order of operations and using the method of substitution and elimination correctly, we can solve systems of linear equations and apply them to real-world problems.