Solve The Following System Of Equations:1. ${$2x + 3y = 6$}$2. { X + 2y = 5$}$
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 using the method of substitution and elimination. We will use the given system of equations as an example to demonstrate the steps involved in solving a system of linear equations.
The System of Equations
The given system of equations is:
- 2x + 3y = 6
- x + 2y = 5
Method of Substitution
The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. We will use this method to solve the given system of equations.
Step 1: Solve the Second Equation for x
We will solve the second equation for x by subtracting 2y from both sides of the equation.
x = 5 - 2y
Step 2: Substitute the Expression for x into the First Equation
We will substitute the expression for x into the first equation by replacing x with (5 - 2y).
2(5 - 2y) + 3y = 6
Step 3: Simplify the Equation
We will simplify the equation by distributing the 2 to the terms inside the parentheses and then combining like terms.
10 - 4y + 3y = 6
Step 4: Combine Like Terms
We will combine the like terms -4y and 3y to get -y.
10 - y = 6
Step 5: Solve for y
We will solve for y by subtracting 10 from both sides of the equation and then multiplying both sides by -1.
-y = -4
y = 4
Step 6: Substitute the Value of y into the Expression for x
We will substitute the value of y into the expression for x to find the value of x.
x = 5 - 2(4)
x = 5 - 8
x = -3
Conclusion
We have solved the given system of equations using the method of substitution. The values of x and y are x = -3 and y = 4, respectively.
Method of Elimination
The method of elimination involves adding or subtracting the equations to eliminate one of the variables. We will use this method to solve the given system of equations.
Step 1: Multiply the First Equation by 2
We will multiply the first equation by 2 to make the coefficients of x in both equations the same.
4x + 6y = 12
Step 2: Multiply the Second Equation by -2
We will multiply the second equation by -2 to make the coefficients of x in both equations the same.
-2x - 4y = -10
Step 3: Add the Two Equations
We will add the two equations to eliminate the variable x.
(4x - 2x) + (6y - 4y) = 12 - 10
2x + 2y = 2
Step 4: Solve for y
We will solve for y by subtracting 2x from both sides of the equation and then dividing both sides by 2.
2y = 2 - 2x
y = 1 - x
Step 5: Substitute the Expression for y into the Second Equation
We will substitute the expression for y into the second equation by replacing y with (1 - x).
x + 2(1 - x) = 5
Step 6: Simplify the Equation
We will simplify the equation by distributing the 2 to the terms inside the parentheses and then combining like terms.
x + 2 - 2x = 5
Step 7: Combine Like Terms
We will combine the like terms x and -2x to get -x.
-x + 2 = 5
Step 8: Solve for x
We will solve for x by subtracting 2 from both sides of the equation and then multiplying both sides by -1.
-x = 3
x = -3
Step 9: Substitute the Value of x into the Expression for y
We will substitute the value of x into the expression for y to find the value of y.
y = 1 - (-3)
y = 1 + 3
y = 4
Conclusion
We have solved the given system of equations using the method of elimination. The values of x and y are x = -3 and y = 4, respectively.
Conclusion
Introduction
In the previous article, we solved a system of two linear equations using the methods 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 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 methods of solving a system of linear equations?
There are two main methods of solving a system of linear equations: substitution and elimination.
Q: What is the method of substitution?
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?
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?
You can choose which method to use based on the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, you can use the method of elimination. If the coefficients of one variable are different in both equations, you can use the method of substitution.
Q: What if I have a system of three or more linear equations?
If you have a system of three or more linear equations, you can use the method of substitution or elimination to solve the system. However, you may need to use a combination of both methods or use a different method such as Gaussian elimination.
Q: What if I have a system of linear equations with no solution?
If you have a system of linear equations with no solution, it means that the equations are inconsistent and there is no value of the variables that satisfies all the equations.
Q: What if I have a system of linear equations with infinitely many solutions?
If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent and there are many values of the variables that satisfy all the equations.
Q: How do I check my work?
To check your work, you can plug the values of the variables back into the original equations to make sure that they are true.
Q: What are some common mistakes to avoid when solving a system of linear equations?
Some common mistakes to avoid when solving a system of linear equations include:
- Not following the order of operations
- Not simplifying the equations
- Not checking the work
- Not using the correct method for the system of equations
Conclusion
In this article, we have answered some frequently asked questions about solving systems of linear equations. We have discussed the methods of substitution and elimination, how to choose which method to use, and how to check your work. We have also discussed some common mistakes to avoid when solving a system of linear equations.
Additional Resources
If you need additional help with solving systems of linear equations, you can try the following resources:
- Online tutorials and videos
- Math textbooks and workbooks
- Online math communities and forums
- Math tutors and instructors
Conclusion
Solving systems of linear equations is an important skill in mathematics and is used in many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving systems of linear equations.