Find The Solution \[$(x, Y)\$\] To These Two Equations Using Elimination:1. \[$4x + 16y = 0\$\]2. \[$4x + 27y = 11\$\]Show All Your Work To Get The Answer.Write The Answer Here \[$\rightarrow\$\] \[$(\ \ ,\ \ )\$\]
Introduction
In mathematics, solving a system of linear equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. There are several methods to solve a system of linear equations, including substitution, elimination, and graphing. In this article, we will focus on the elimination method, which involves adding or subtracting equations to eliminate one of the variables.
Understanding the Elimination Method
The elimination method is a powerful technique for solving systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the remaining variable. The key to the elimination method is to find a way to eliminate one of the variables by making the coefficients of that variable the same in both equations, but with opposite signs.
Step 1: Write Down the Equations
To begin solving the system of linear equations using the elimination method, we need to write down the equations. In this case, we have two equations:
- 4x + 16y = 0
- 4x + 27y = 11
Step 2: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to make the coefficients of that variable the same in both equations, but with opposite signs. We can do this by multiplying the equations by necessary multiples.
Let's multiply the first equation by 27 and the second equation by 16:
- (4x + 16y = 0) × 27 => 108x + 432y = 0
- (4x + 27y = 11) × 16 => 64x + 432y = 176
Step 3: Subtract the Second Equation from the First Equation
Now that we have the equations with the same coefficient for y, but with opposite signs, we can subtract the second equation from the first equation to eliminate the variable y.
(108x + 432y = 0) - (64x + 432y = 176)
This simplifies to:
44x = -176
Step 4: Solve for x
Now that we have the equation 44x = -176, we can solve for x by dividing both sides of the equation by 44.
x = -176 / 44
x = -4
Step 5: Substitute 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 use the first equation:
4x + 16y = 0
Substituting x = -4, we get:
4(-4) + 16y = 0
This simplifies to:
-16 + 16y = 0
Step 6: Solve for y
Now that we have the equation -16 + 16y = 0, we can solve for y by adding 16 to both sides of the equation and then dividing both sides by 16.
16y = 16
y = 16 / 16
y = 1
Conclusion
In this article, we used the elimination method to solve a system of linear equations. We started by writing down the equations, then multiplied the equations by necessary multiples to make the coefficients of the variable y the same in both equations, but with opposite signs. We then subtracted the second equation from the first equation to eliminate the variable y, and solved for x. Finally, we substituted x into one of the original equations to solve for y.
The solution to the system of linear equations is:
(x, y) = (-4, 1)
This means that the values of x and y that satisfy both equations are x = -4 and y = 1.
Introduction
In our previous article, we discussed how to solve a system of linear equations using the elimination method. This method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the remaining variable. In this article, we will answer some frequently asked questions about solving systems of linear equations using the elimination method.
Q: What is the elimination method?
A: The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method involves making the coefficients of the variable to be eliminated the same in both equations, but with opposite signs.
Q: How do I know which variable to eliminate first?
A: To determine which variable to eliminate first, you need to look at the coefficients of the variables in both equations. If the coefficients of one variable are the same in both equations, but with opposite signs, you can eliminate that variable first. If not, you need to multiply one or both equations by necessary multiples to make the coefficients of one variable the same in both equations, but with opposite signs.
Q: What if I have two equations with the same coefficients for the variable to be eliminated?
A: If you have two equations with the same coefficients for the variable to be eliminated, you can add or subtract the equations to eliminate that variable. For example, if you have two equations:
2x + 3y = 5
2x + 3y = 7
You can add the equations to eliminate the variable x:
(2x + 3y = 5) + (2x + 3y = 7)
This simplifies to:
4x + 6y = 12
Q: What if I have two equations with different coefficients for the variable to be eliminated?
A: If you have two equations with different coefficients for the variable to be eliminated, you need to multiply one or both equations by necessary multiples to make the coefficients of one variable the same in both equations, but with opposite signs. For example, if you have two equations:
2x + 3y = 5
4x + 5y = 7
You can multiply the first equation by 2 and the second equation by 3 to make the coefficients of x the same in both equations, but with opposite signs:
(2x + 3y = 5) × 2 => 4x + 6y = 10
(4x + 5y = 7) × 3 => 12x + 15y = 21
Q: How do I know if I have made a mistake in the elimination method?
A: To check if you have made a mistake in the elimination method, you need to substitute the values of x and y back into the original equations to see if they are true. If the values of x and y do not satisfy both equations, you need to recheck your work and try again.
Q: Can I use the elimination method to solve systems of linear equations with more than two variables?
A: Yes, you can use the elimination method to solve systems of linear equations with more than two variables. However, you need to be careful when eliminating variables to make sure that you do not eliminate a variable that is not present in both equations.
Conclusion
In this article, we answered some frequently asked questions about solving systems of linear equations using the elimination method. We discussed how to determine which variable to eliminate first, how to handle equations with the same coefficients for the variable to be eliminated, and how to check if you have made a mistake in the elimination method. We also discussed how to use the elimination method to solve systems of linear equations with more than two variables.
By following these tips and techniques, you can use the elimination method to solve systems of linear equations with confidence and accuracy.