Solve The Following System Of Equations:${ \begin{array}{l} -10x + Y = 4 \ -6x - Y = 12 \end{array} }$
Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.
What is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations that involve the same set of variables. Each equation in the system 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 the variables.
Example: A System of Two Linear Equations
Consider the following system of two linear equations:
{ \begin{array}{l} -10x + y = 4 \\ -6x - y = 12 \end{array} \}
This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations.
Method 1: Substitution Method
One way to solve a system of linear equations is to use the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.
Step 1: Solve the First Equation for y
Let's solve the first equation for y:
-10x + y = 4
Add 10x to both sides of the equation:
y = 10x + 4
Step 2: Substitute the Expression for y into the Second Equation
Now, substitute the expression for y into the second equation:
-6x - (10x + 4) = 12
Combine like terms:
-16x - 4 = 12
Add 4 to both sides of the equation:
-16x = 16
Divide both sides of the equation by -16:
x = -1
Step 3: Find the Value of y
Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:
-10x + y = 4
Substitute x = -1 into the equation:
-10(-1) + y = 4
Simplify the equation:
10 + y = 4
Subtract 10 from both sides of the equation:
y = -6
Conclusion
In this article, we have solved a system of two linear equations using the substitution method. We first solved one equation for one variable and then substituted that expression into the other equation. This allowed us to find the values of both variables that satisfy both equations.
Method 2: Elimination Method
Another way to solve a system of linear equations is to use the elimination method. This method involves adding or subtracting the equations in the system to eliminate one of the variables.
Step 1: Add the Two Equations
Let's add the two equations in the system:
(-10x + y) + (-6x - y) = 4 + 12
Combine like terms:
-16x = 16
Divide both sides of the equation by -16:
x = -1
Step 2: Find the Value of y
Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:
-10x + y = 4
Substitute x = -1 into the equation:
-10(-1) + y = 4
Simplify the equation:
10 + y = 4
Subtract 10 from both sides of the equation:
y = -6
Conclusion
In this article, we have solved a system of two linear equations using the elimination method. We first added the two equations in the system to eliminate one of the variables. This allowed us to find the values of both variables that satisfy both equations.
Conclusion
In this article, we have discussed two methods for solving a system of linear equations: the substitution method and the elimination method. We have used these methods to solve a system of two linear equations with two variables. By following these steps, you can solve systems of linear equations with two or more variables.
Tips and Tricks
- Make sure to check your work by plugging the values of the variables back into the original equations.
- Use a graphing calculator or a computer algebra system to check your work and to visualize the solution.
- Practice solving systems of linear equations with different numbers of variables and equations.
References
- [1] "Linear Algebra and Its Applications" by Gilbert Strang
- [2] "Introduction to Linear Algebra" by Jim Hefferon
- [3] "Linear Algebra: A Modern Introduction" by David Poole
Final Thoughts
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 the same set of variables. Each equation in the system 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 the variables.
Q: How do I know if a system of linear equations has a solution?
A: A system of linear equations has a solution if and only if the two equations are consistent, meaning that they do not contradict each other. If the two equations are inconsistent, then the system has no solution.
Q: What are the two main methods for solving systems of linear equations?
A: The two main methods for solving systems of linear equations are the substitution method and the elimination method.
- The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
- The elimination method involves adding or subtracting the equations in the system to eliminate one of the variables.
Q: How do I choose between the substitution method and the elimination method?
A: You can choose between the substitution method and the elimination method based on the form of the equations in the system. If one equation is already solved for one variable, then the substitution method may be easier to use. If the equations are not in a form that is easily solvable for one variable, then the elimination method may be easier to use.
Q: What if I have a system of linear equations with three or more variables?
A: If you have a system of linear equations with three or more variables, then you can use the same methods that you would use for a system of two variables. However, the process may be more complicated and may require the use of matrices or other advanced techniques.
Q: How do I check my work when solving a system of linear equations?
A: To check your work when solving a system of linear equations, you can plug the values of the variables back into the original equations and make sure that they are true. You can also use a graphing calculator or a computer algebra system to check your work and to visualize the solution.
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 your work
- Not using the correct method for the form of the equations
- Not following the steps of the method correctly
- Not plugging the values of the variables back into the original equations to check your work
Q: How can I practice solving systems of linear equations?
A: You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear equations on your own, using a graphing calculator or a computer algebra system to check your work.
Q: What are some real-world applications of solving systems of linear equations?
A: Solving systems of linear equations has many real-world applications, including:
- Physics and engineering: Solving systems of linear equations is used to model and solve problems in physics and engineering, such as the motion of objects and the behavior of electrical circuits.
- Economics: Solving systems of linear equations is used to model and solve problems in economics, such as the behavior of supply and demand and the impact of taxes on the economy.
- Computer science: Solving systems of linear equations is used in computer science to solve problems in computer graphics, game development, and other areas.
Conclusion
Solving systems of linear equations is an important skill in mathematics and science. By following the steps outlined in this article and practicing solving systems of linear equations, you can develop the skills and knowledge you need to solve problems in a variety of fields. Remember to check your work and to use the correct method for the form of the equations.