Use The Elimination Method To Solve The System Of Equations. Choose The Correct Ordered Pair.$\[ \begin{array}{l} 2x + 4y = 16 \\ 2x - 4y = 0 \end{array} \\]A. \[$(2, -4)\$\] B. \[$(2, 4)\$\] C. \[$(4, -2)\$\] D.
Introduction
In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. The elimination method is one of the most common techniques used to solve systems of equations. This method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. In this article, we will use the elimination method to solve a system of two linear equations and choose the correct ordered pair.
The System of Equations
The given system of equations is:
{ \begin{array}{l} 2x + 4y = 16 \\ 2x - 4y = 0 \end{array} \}
Our goal is to solve this system of equations using the elimination method and choose the correct ordered pair.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to make the coefficients of either x or y 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 1 and the second equation by 1. This will give us:
{ \begin{array}{l} 2x + 4y = 16 \\ 2x - 4y = 0 \end{array} \}
Step 2: Add or Subtract the Equations
Now that we have the equations with the same coefficients, we can add or subtract them to eliminate one of the variables. Let's add the two equations to eliminate the variable x.
Simplifying the equation, we get:
Step 3: Solve for the Variable
Now that we have the equation with only one variable, we can solve for that variable. Let's solve for x.
Dividing both sides by 4, we get:
Step 4: Substitute the Value of the Variable 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 the other variable. Let's substitute x = 4 into the first equation.
Simplifying the equation, we get:
Subtracting 8 from both sides, we get:
Dividing both sides by 4, we get:
Step 5: Write the Solution as an Ordered Pair
Now that we have the values of x and y, we can write the solution as an ordered pair.
Conclusion
In this article, we used the elimination method to solve a system of two linear equations and chose the correct ordered pair. We multiplied the equations by necessary multiples, added or subtracted the equations to eliminate one of the variables, solved for the variable, substituted the value of the variable into one of the original equations, and wrote the solution as an ordered pair.
Answer
The correct ordered pair is:
This is option C.
Discussion
The elimination method is a powerful technique used to solve systems of equations. It involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. In this article, we used the elimination method to solve a system of two linear equations and chose the correct ordered pair.
The elimination method can be used to solve systems of equations with two or more variables. It is a useful technique to have in your mathematical toolkit, especially when dealing with systems of equations.
Example Problems
Here are some example problems that you can try to practice the elimination method:
- Solve the system of equations using the elimination method:
{ \begin{array}{l} x + 2y = 6 \\ 3x - 2y = 2 \end{array} \}
- Solve the system of equations using the elimination method:
{ \begin{array}{l} 2x + 3y = 12 \\ x - 2y = -3 \end{array} \}
- Solve the system of equations using the elimination method:
{ \begin{array}{l} x + y = 5 \\ 2x - 3y = -1 \end{array} \}
Tips and Tricks
Here are some tips and tricks to help you solve systems of equations using the elimination method:
- Make sure to multiply the equations by necessary multiples to make the coefficients of either x or y the same in both equations, but with opposite signs.
- Add or subtract the equations to eliminate one of the variables.
- Solve for the variable by dividing both sides of the equation by the coefficient of the variable.
- Substitute the value of the variable into one of the original equations to solve for the other variable.
- Write the solution as an ordered pair.
Q: What is the elimination method?
A: The elimination method is a technique used to solve systems of equations by adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable.
Q: How do I know which variable to eliminate?
A: To determine which variable to eliminate, 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 by adding or subtracting the equations.
Q: What if the coefficients of the variables are not the same in both equations?
A: If the coefficients of the variables are not the same in both equations, you can multiply the equations by necessary multiples to make the coefficients of either x or y the same in both equations, but with opposite signs.
Q: How do I add or subtract the equations to eliminate one of the variables?
A: To add or subtract the equations, simply add or subtract the corresponding terms of the two equations. For example, if you have the equations 2x + 4y = 16 and 2x - 4y = 0, you can add the two equations to eliminate the variable y.
Q: What if I get a fraction or a decimal when I add or subtract the equations?
A: If you get a fraction or a decimal when you add or subtract the equations, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fraction or decimal.
Q: How do I solve for the variable after eliminating it?
A: After eliminating one of the variables, you can solve for the other variable by dividing both sides of the equation by the coefficient of the variable.
Q: Can I use the elimination method to solve systems of equations with more than two variables?
A: Yes, you can use the elimination method to solve systems of equations with more than two variables. However, you will need to eliminate one variable at a time, using the same steps as before.
Q: What if I get a system of equations with no solution?
A: If you get a system of equations with no solution, it means that the equations are inconsistent, and there is no value of the variables that can satisfy both equations.
Q: What if I get a system of equations with infinitely many solutions?
A: If you get a system of equations with infinitely many solutions, it means that the equations are dependent, and there are many values of the variables that can satisfy both equations.
Q: Can I use the elimination method to solve systems of equations with non-linear equations?
A: No, the elimination method is only used to solve systems of linear equations. If you have a system of non-linear equations, you will need to use a different method, such as substitution or graphing.
Q: What are some common mistakes to avoid when using the elimination method?
A: Some common mistakes to avoid when using the elimination method include:
- Not multiplying the equations by necessary multiples to make the coefficients of either x or y the same in both equations, but with opposite signs.
- Not adding or subtracting the equations correctly.
- Not solving for the variable after eliminating it.
- Not checking for fractions or decimals when adding or subtracting the equations.
By following these tips and avoiding common mistakes, you can become proficient in using the elimination method to solve systems of equations.