Solve The System Using Elimination:${ \begin{align*} 2x + 3y &= 8 \ 5x + 6y &= 17 \ \end{align*} }$([?], \square)
Introduction
In this article, we will explore the method of elimination to solve a system of linear equations. The elimination method is a powerful technique used to solve systems of linear equations by adding or subtracting equations to eliminate variables. This method is particularly useful when the coefficients of the variables in the equations are relatively simple.
What is the Elimination Method?
The elimination method involves adding or subtracting equations to eliminate one of the variables. This is done by multiplying the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same. Once the coefficients are the same, we can add or subtract the equations to eliminate the variable.
Step 1: Write Down the Equations
The given system of linear equations is:
{ \begin{align*} 2x + 3y &= 8 \\ 5x + 6y &= 17 \\ \end{align*} \}
Step 2: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same. Let's multiply the first equation by 5 and the second equation by 2.
{ \begin{align*} 10x + 15y &= 40 \\ 10x + 12y &= 34 \\ \end{align*} \}
Step 3: Subtract the Second Equation from the First Equation
Now that the coefficients of the variable x are the same, we can subtract the second equation from the first equation to eliminate the variable x.
{ \begin{align*} (10x + 15y) - (10x + 12y) &= 40 - 34 \\ 3y &= 6 \\ \end{align*} \}
Step 4: Solve for y
Now that we have eliminated the variable x, we can solve for y by dividing both sides of the equation by 3.
{ \begin{align*} y &= \frac{6}{3} \\ y &= 2 \\ \end{align*} \}
Step 5: Substitute the Value of y into One of the Original Equations
Now that we have the value of y, we can substitute it into one of the original equations to solve for x. Let's substitute the value of y into the first equation.
{ \begin{align*} 2x + 3(2) &= 8 \\ 2x + 6 &= 8 \\ 2x &= 2 \\ x &= 1 \\ \end{align*} \}
Conclusion
In this article, we have used the elimination method to solve a system of linear equations. We have multiplied the equations by necessary multiples, subtracted the second equation from the first equation, solved for y, and substituted the value of y into one of the original equations to solve for x. The final solution is x = 1 and y = 2.
Example Problems
Here are some example problems that you can try to practice the elimination method:
- Solve the system of linear equations using the elimination method:
{ \begin{align*} x + 2y &= 6 \\ 3x + 4y &= 14 \\ \end{align*} \}
- Solve the system of linear equations using the elimination method:
{ \begin{align*} 2x + 3y &= 12 \\ x + 2y &= 5 \\ \end{align*} \}
Tips and Tricks
Here are some tips and tricks that you can use to solve systems of linear equations using the elimination method:
- Multiply the equations by necessary multiples to eliminate one of the variables.
- Add or subtract the equations to eliminate the variable.
- Solve for the variable that is eliminated.
- Substitute the value of the eliminated variable into one of the original equations to solve for the other variable.
- Check your solution by plugging it back into the original equations.
Conclusion
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 variables.
Q: How do I know which variable to eliminate first?
A: To determine which variable to eliminate first, look for the variables that have the same coefficient in both equations. Eliminate the variable with the smaller coefficient first.
Q: Can I eliminate both variables at the same time?
A: No, you cannot eliminate both variables at the same time. The elimination method involves eliminating one variable at a time.
Q: What if the coefficients of the variables are not the same?
A: If the coefficients of the variables are not the same, you can multiply the equations by necessary multiples to make the coefficients the same.
Q: Can I use the elimination method to solve a system of linear equations with three variables?
A: Yes, you can use the elimination method to solve a system of linear equations with three variables. However, you will need to eliminate two variables at a time.
Q: How do I know if the system of linear equations has a unique solution, no solution, or infinitely many solutions?
A: To determine if the system of linear equations has a unique solution, no solution, or infinitely many solutions, check the following:
- If the system has a unique solution, the equations will be consistent and the solution will be a single point.
- If the system has no solution, the equations will be inconsistent and there will be no solution.
- If the system has infinitely many solutions, the equations will be consistent and the solution will be a line or a plane.
Q: Can I use the elimination method to solve a system of linear equations with fractions?
A: Yes, you can use the elimination method to solve a system of linear equations with fractions. However, you will need to multiply the equations by necessary multiples to eliminate the fractions.
Q: How do I check my solution to make sure it is correct?
A: To check your solution, plug the values of the variables back into the original equations and make sure they are true.
Q: Can I use the elimination method to solve a system of linear equations with decimals?
A: Yes, you can use the elimination method to solve a system of linear equations with decimals. However, you will need to round the decimals to a reasonable number of decimal places to make the calculations easier.
Q: How do I know if the system of linear equations is consistent or inconsistent?
A: To determine if the system of linear equations is consistent or inconsistent, check the following:
- If the system is consistent, the equations will have a solution.
- If the system is inconsistent, the equations will have no solution.
Q: Can I use the elimination method to solve a system of linear equations with negative coefficients?
A: Yes, you can use the elimination method to solve a system of linear equations with negative coefficients. However, you will need to multiply the equations by necessary multiples to make the coefficients positive.
Conclusion
In conclusion, the elimination method is a powerful technique used to solve systems of linear equations. By understanding the basics of the elimination method, you can solve systems of linear equations with ease. Remember to check your solution to make sure it is correct and to use the elimination method to solve systems of linear equations with fractions, decimals, and negative coefficients.