Solve The System Of Equations:${ \begin{array}{l} 4x + 3y = 18 \ 4x - 2y = 8 \end{array} }$
Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that are to be solved simultaneously. These equations are linear because they are in the form of a straight line, and they are solved simultaneously because we need to find the values of the variables that satisfy all the equations at the same time. 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 are to be solved simultaneously. Each equation in the system is a linear equation, which means that it is in the form of a straight line. The general form of a linear equation is:
ax + by = c
where a, b, and c are constants, and x and y are variables.
Example of a System of Linear Equations
The following is an example of a system of two linear equations with two variables:
4x + 3y = 18
4x - 2y = 8
In this system, we have two linear equations with two variables, x and y. We need to find the values of x and y that satisfy both equations at the same time.
Methods for Solving a System of Linear Equations
There are several methods for solving a system 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 equations to eliminate one variable.
- Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
Solving the System of Linear Equations using the Substitution Method
To solve the system of linear equations using the substitution method, we will first solve one equation for one variable. Let's solve the first equation for x:
4x + 3y = 18
4x = 18 - 3y
x = (18 - 3y) / 4
Now, we will substitute this expression for x into the second equation:
4x - 2y = 8
4((18 - 3y) / 4) - 2y = 8
18 - 3y - 2y = 8
18 - 5y = 8
-5y = -10
y = 2
Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x. Let's substitute y = 2 into the first equation:
4x + 3y = 18
4x + 3(2) = 18
4x + 6 = 18
4x = 12
x = 3
Therefore, the solution to the system of linear equations is x = 3 and y = 2.
Solving the System of Linear Equations using the Elimination Method
To solve the system of linear equations using the elimination method, we will add or subtract the equations to eliminate one variable. Let's add the two equations:
4x + 3y = 18
4x - 2y = 8
(4x + 3y) + (4x - 2y) = 18 + 8
8x + y = 26
Now, we can solve for x:
8x = 26 - y
x = (26 - y) / 8
Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y. Let's substitute x = (26 - y) / 8 into the first equation:
4x + 3y = 18
4((26 - y) / 8) + 3y = 18
(26 - y) + 3y = 18
26 + 2y = 18
2y = -8
y = -4
Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x. Let's substitute y = -4 into the first equation:
4x + 3y = 18
4x + 3(-4) = 18
4x - 12 = 18
4x = 30
x = 7.5
Therefore, the solution to the system of linear equations is x = 7.5 and y = -4.
Conclusion
In this article, we have discussed how to solve a system of linear equations using the substitution method and the elimination method. We have also provided examples of how to solve a system of linear equations using these methods. 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 to eliminate one variable. By following these methods, we can solve systems of linear equations and find the values of the variables that satisfy all the equations at the same time.
References
- "Linear Algebra and Its Applications" by Gilbert Strang
- "Introduction to Linear Algebra" by Gilbert Strang
- "Solving Systems of Linear Equations" by Math Open Reference
Further Reading
- "Linear Algebra" by Khan Academy
- "Solving Systems of Linear Equations" by Mathway
- "Linear Algebra and Its Applications" by Wolfram MathWorld
Solving a System of Linear Equations: Q&A =====================================
Introduction
In our previous article, we discussed how to solve a system of linear equations using the substitution method and the elimination method. In this article, we will provide a Q&A section to help you better understand the concepts and methods discussed earlier.
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 to be solved simultaneously. Each equation in the system is a linear equation, which means that it is in the form of a straight line.
Q: What are the methods for solving a system of linear equations?
A: There are several methods for solving a system 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 equations to eliminate one variable.
- 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 specific system of equations and the variables involved. If the equations are simple and easy to solve, the substitution method may be the best choice. If the equations are more complex, the elimination method may be more suitable.
Q: What is the substitution method?
A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one of the variables is easily solved for.
Q: What is the elimination method?
A: The elimination method involves adding or subtracting the equations to eliminate one variable. This method is useful when the coefficients of the variables are the same in both equations.
Q: How do I solve a system of linear equations using the substitution method?
A: To solve a system of linear equations using the substitution method, follow these steps:
- Solve one equation for one variable.
- Substitute that expression into the other equation.
- Solve for the remaining variable.
Q: How do I solve a system of linear equations using the elimination method?
A: To solve a system of linear equations using the elimination method, follow these steps:
- Add or subtract the equations to eliminate one variable.
- Solve for the remaining variable.
- Substitute that expression into one of the original equations to find the value of the other variable.
Q: What are some common mistakes to avoid when solving a system of linear equations?
A: Some common mistakes to avoid when solving a system of linear equations include:
- Not checking the solution: Make sure to check the solution by substituting it back into both original equations.
- Not following the correct order of operations: Make sure to follow the correct order of operations when solving the equations.
- Not using the correct method: Make sure to use the correct method for the specific system of equations.
Q: How do I know if a system of linear equations has a solution?
A: A system of linear equations has a solution if the equations are consistent and the variables are related in a way that allows for a solution. If the equations are inconsistent or the variables are unrelated, there may not be a solution.
Conclusion
In this article, we have provided a Q&A section to help you better understand the concepts and methods discussed earlier. We hope that this article has been helpful in answering your questions and providing a better understanding of how to solve a system of linear equations.
References
- "Linear Algebra and Its Applications" by Gilbert Strang
- "Introduction to Linear Algebra" by Gilbert Strang
- "Solving Systems of Linear Equations" by Math Open Reference
Further Reading
- "Linear Algebra" by Khan Academy
- "Solving Systems of Linear Equations" by Mathway
- "Linear Algebra and Its Applications" by Wolfram MathWorld