Find The Solution To The System Of Equations Given Below Using Elimination.${ \begin{array}{l} 3x + 2y = -3 \ 9x + 4y = 3 \end{array} }$
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Introduction
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 is particularly useful when the coefficients of the variables in the equations are relatively simple and can be easily manipulated. In this article, we will use the elimination method to solve the system of equations given below.
The System of Equations
The system of equations we will be solving is:
{ \begin{array}{l} 3x + 2y = -3 \\ 9x + 4y = 3 \end{array} \}
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. We can do this by multiplying the equations by necessary multiples. Let's multiply the first equation by 3 and the second equation by 1.
{ \begin{array}{l} 9x + 6y = -9 \\ 9x + 4y = 3 \end{array} \}
Step 2: Subtract the Second Equation from the First Equation
Now that we have the same coefficient for x in both equations, we can subtract the second equation from the first equation to eliminate x.
{ \begin{array}{l} (9x + 6y) - (9x + 4y) = -9 - 3 \\ 2y = -12 \end{array} \}
Step 3: Solve for y
Now that we have eliminated x, we can solve for y by dividing both sides of the equation by 2.
{ \begin{array}{l} y = -12/2 \\ y = -6 \end{array} \}
Step 4: 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 y = -6 into the first equation.
{ \begin{array}{l} 3x + 2(-6) = -3 \\ 3x - 12 = -3 \end{array} \}
Step 5: Solve for x
Now that we have the value of y substituted into the equation, we can solve for x by adding 12 to both sides of the equation and then dividing both sides by 3.
{ \begin{array}{l} 3x = -3 + 12 \\ 3x = 9 \\ x = 9/3 \\ x = 3 \end{array} \}
Conclusion
In this article, we used the elimination method to solve the system of equations given below.
{ \begin{array}{l} 3x + 2y = -3 \\ 9x + 4y = 3 \end{array} \}
We multiplied the equations by necessary multiples, subtracted the second equation from the first equation, solved for y, substituted the value of y into one of the original equations, and finally solved for x. The solution to the system of equations is x = 3 and y = -6.
Example Problems
Problem 1
Solve the system of equations using the elimination method.
{ \begin{array}{l} 2x + 3y = 7 \\ 4x + 6y = 15 \end{array} \}
Solution
To solve this system of equations, we can multiply the first equation by 2 and the second equation by 1.
{ \begin{array}{l} 4x + 6y = 14 \\ 4x + 6y = 15 \end{array} \}
Now, we can subtract the first equation from the second equation to eliminate x.
{ \begin{array}{l} (4x + 6y) - (4x + 6y) = 15 - 14 \\ 0 = 1 \end{array} \}
This is a contradiction, which means that the system of equations has no solution.
Problem 2
Solve the system of equations using the elimination method.
{ \begin{array}{l} x + 2y = 6 \\ 2x + 4y = 12 \end{array} \}
Solution
To solve this system of equations, we can multiply the first equation by 2 and the second equation by 1.
{ \begin{array}{l} 2x + 4y = 12 \\ 2x + 4y = 12 \end{array} \}
Now, we can subtract the first equation from the second equation to eliminate x.
{ \begin{array}{l} (2x + 4y) - (2x + 4y) = 12 - 12 \\ 0 = 0 \end{array} \}
This is an identity, which means that the system of equations is dependent and has infinitely many solutions.
Tips and Tricks
- When 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.
- When subtracting one equation from another, make sure to subtract the corresponding terms.
- When solving for x or y, make sure to isolate the variable by adding or subtracting the same value to both sides of the equation.
- When the system of equations has no solution, it means that the equations are inconsistent and cannot be solved simultaneously.
- When the system of equations is dependent, it means that the equations are equivalent and have infinitely many solutions.
Conclusion
In this article, we used the elimination method to solve systems of linear equations. We multiplied the equations by necessary multiples, subtracted the second equation from the first equation, solved for y, substituted the value of y into one of the original equations, and finally solved for x. We also provided example problems and tips and tricks to help you understand the elimination method better.
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Q1: What is the elimination method?
A1: The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.
Q2: How do I choose which variable to eliminate?
A2: You can choose either variable to eliminate, but it's often easier to eliminate the variable with the larger coefficient.
Q3: What if the coefficients of the variables are not the same in both equations?
A3: In this case, you can multiply one or both of the equations by necessary multiples to make the coefficients of the variables the same.
Q4: What if I get a contradiction when subtracting one equation from another?
A4: If you get a contradiction, it means that the system of equations has no solution.
Q5: What if I get an identity when subtracting one equation from another?
A5: If you get an identity, it means that the system of equations is dependent and has infinitely many solutions.
Q6: Can I use the elimination method to solve systems of equations with more than two variables?
A6: Yes, you can use the elimination method to solve systems of equations with more than two variables, but it may be more complicated.
Q7: What if I'm not sure which method to use to solve a system of equations?
A7: If you're not sure which method to use, you can try using the elimination method first. If it doesn't work, you can try using the substitution method or graphing.
Q8: Can I use a calculator to solve systems of equations using the elimination method?
A8: Yes, you can use a calculator to solve systems of equations using the elimination method, but it's often easier to do it by hand.
Q9: What if I make a mistake when solving a system of equations using the elimination method?
A9: If you make a mistake, you can try reworking the problem or using a different method to solve it.
Q10: Is the elimination method the only method to solve systems of equations?
A10: No, there are several other methods to solve systems of equations, including the substitution method, graphing, and matrix methods.
Conclusion
In this article, we answered frequently asked questions about solving systems of equations using the elimination method. We covered topics such as choosing which variable to eliminate, multiplying equations by necessary multiples, and dealing with contradictions and identities. We also provided tips and tricks to help you understand the elimination method better.
Example Problems
Problem 1
Solve the system of equations using the elimination method.
{ \begin{array}{l} 2x + 3y = 7 \\ 4x + 6y = 15 \end{array} \}
Solution
To solve this system of equations, we can multiply the first equation by 2 and the second equation by 1.
{ \begin{array}{l} 4x + 6y = 14 \\ 4x + 6y = 15 \end{array} \}
Now, we can subtract the first equation from the second equation to eliminate x.
{ \begin{array}{l} (4x + 6y) - (4x + 6y) = 15 - 14 \\ 0 = 1 \end{array} \}
This is a contradiction, which means that the system of equations has no solution.
Problem 2
Solve the system of equations using the elimination method.
{ \begin{array}{l} x + 2y = 6 \\ 2x + 4y = 12 \end{array} \}
Solution
To solve this system of equations, we can multiply the first equation by 2 and the second equation by 1.
{ \begin{array}{l} 2x + 4y = 12 \\ 2x + 4y = 12 \end{array} \}
Now, we can subtract the first equation from the second equation to eliminate x.
{ \begin{array}{l} (2x + 4y) - (2x + 4y) = 12 - 12 \\ 0 = 0 \end{array} \}
This is an identity, which means that the system of equations is dependent and has infinitely many solutions.
Tips and Tricks
- When 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.
- When subtracting one equation from another, make sure to subtract the corresponding terms.
- When solving for x or y, make sure to isolate the variable by adding or subtracting the same value to both sides of the equation.
- When the system of equations has no solution, it means that the equations are inconsistent and cannot be solved simultaneously.
- When the system of equations is dependent, it means that the equations are equivalent and have infinitely many solutions.