Solve The Following System Of Equations:$\[ \begin{array}{l} -4x - 2y = 14 \\ -10x + 7y = -25 \end{array} \\]
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 to the system.
The System of Equations
The system of equations we will be solving is:
{ \begin{array}{l} -4x - 2y = 14 \\ -10x + 7y = -25 \end{array} \}
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of either x or y in both equations are the same. Let's multiply the first equation by 5 and the second equation by 2.
{ \begin{array}{l} -20x - 10y = 70 \\ -20x + 14y = -50 \end{array} \}
Step 2: Subtract the Second Equation from the First Equation
Now, we can subtract the second equation from the first equation to eliminate the variable x.
{ \begin{array}{l} -20x - 10y - (-20x + 14y) = 70 - (-50) \\ -24y = 120 \end{array} \}
Step 3: Solve for y
Now, we can solve for y by dividing both sides of the equation by -24.
{ \begin{array}{l} y = -120/24 \\ y = -5 \end{array} \}
Step 4: Substitute the Value of y into One of the Original Equations
Now that we have the value of y, we can substitute it into one of the original equations to solve for x. Let's substitute y = -5 into the first equation.
{ \begin{array}{l} -4x - 2(-5) = 14 \\ -4x + 10 = 14 \end{array} \}
Step 5: Solve for x
Now, we can solve for x by subtracting 10 from both sides of the equation and then dividing both sides by -4.
{ \begin{array}{l} -4x = 14 - 10 \\ -4x = 4 \\ x = -4/4 \\ x = -1 \end{array} \}
Conclusion
In this article, we solved a system of two linear equations with two variables using the method of substitution and elimination. We first multiplied the equations by necessary multiples to eliminate one of the variables, then subtracted the second equation from the first equation to eliminate the variable x. We solved for y by dividing both sides of the equation by -24, and then substituted the value of y into one of the original equations to solve for x. The solution to the system is x = -1 and y = -5.
Real-World Applications
Solving systems of linear equations has many real-world applications, including:
- Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.
- Economics: Systems of linear equations are used to model economic systems, such as supply and demand, and to make predictions about the behavior of economic variables.
- Computer Science: Systems of linear equations are used in computer graphics, computer vision, and machine learning to solve problems such as image recognition, object detection, and natural language processing.
Tips and Tricks
Here are some tips and tricks for solving systems of linear equations:
- Use the method of substitution and elimination: This method is the most efficient way to solve systems of linear equations.
- Check your work: Always check your work by plugging the values of x and y back into the original equations to make sure they are true.
- Use a graphing calculator: A graphing calculator can be a useful tool for visualizing the solution to a system of linear equations.
Common Mistakes
Here are some common mistakes to avoid when solving systems of linear equations:
- Not checking your work: Always check your work by plugging the values of x and y back into the original equations to make sure they are true.
- Not using the method of substitution and elimination: This method is the most efficient way to solve systems of linear equations.
- Not using a graphing calculator: A graphing calculator can be a useful tool for visualizing the solution to a system of linear equations.
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 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.
Q: What is the method of substitution?
A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.
Q: What is the method of elimination?
A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables.
Q: How do I choose which method to use?
A: The choice of method depends on the coefficients of the variables in the equations. If the coefficients are the same, use the method of elimination. If the coefficients are different, use the method of substitution.
Q: What are some common mistakes to avoid when solving systems of linear equations?
A: Some common mistakes to avoid include:
- Not checking your work
- Not using the method of substitution and elimination
- Not using a graphing calculator
- Not following the steps outlined in the solution
Q: How do I check my work?
A: To check your work, plug the values of x and y back into the original equations to make sure they are true.
Q: What are some real-world applications of solving systems of linear equations?
A: Some real-world applications of solving systems of linear equations include:
- Physics and engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.
- Economics: Systems of linear equations are used to model economic systems, such as supply and demand, and to make predictions about the behavior of economic variables.
- Computer science: Systems of linear equations are used in computer graphics, computer vision, and machine learning to solve problems such as image recognition, object detection, and natural language processing.
Q: How do I use a graphing calculator to solve systems of linear equations?
A: To use a graphing calculator to solve systems of linear equations, follow these steps:
- Enter the equations into the calculator.
- Use the calculator to graph the equations.
- Use the calculator to find the intersection point of the two lines.
- Use the intersection point to find the values of x and y.
Q: What are some tips for solving systems of linear equations?
A: Some tips for solving systems of linear equations include:
- Use the method of substitution and elimination.
- Check your work.
- Use a graphing calculator.
- Follow the steps outlined in the solution.
Q: How do I solve a system of linear equations with three variables?
A: To solve a system of linear equations with three variables, use the method of substitution and elimination. First, solve one equation for one variable and then substitute that expression into the other equations. Then, use the method of elimination to eliminate one of the variables.
Q: What are some common types of systems of linear equations?
A: Some common types of systems of linear equations include:
- Homogeneous systems: Systems of linear equations where the constant term is zero.
- Nonhomogeneous systems: Systems of linear equations where the constant term is not zero.
- Consistent systems: Systems of linear equations where the solution is unique.
- Inconsistent systems: Systems of linear equations where the solution is not unique.
Q: How do I determine if a system of linear equations is consistent or inconsistent?
A: To determine if a system of linear equations is consistent or inconsistent, use the following steps:
- Check if the system has a unique solution.
- Check if the system has no solution.
- Check if the system has infinitely many solutions.
If the system has a unique solution, it is consistent. If the system has no solution, it is inconsistent. If the system has infinitely many solutions, it is inconsistent.