Solve Using Elimination.$\[ \begin{array}{l} -7x + 6y = -13 \\ -7x + 10y = 11 \end{array} \\]
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 the same, but the constants are different. In this article, we will discuss how to solve a system of linear equations using the elimination method.
Understanding the Problem
The given problem is a system of two linear equations with two variables, x and y. The equations are:
{ \begin{array}{l} -7x + 6y = -13 \\ -7x + 10y = 11 \end{array} \}
Our goal is to find the values of x and y that satisfy both equations.
Step 1: Write Down the Equations
The first step is to write down the given equations:
{ \begin{array}{l} -7x + 6y = -13 \\ -7x + 10y = 11 \end{array} \}
Step 2: Identify the Coefficients
The next step is to identify the coefficients of x and y in both equations. In the first equation, the coefficient of x is -7 and the coefficient of y is 6. In the second equation, the coefficient of x is also -7 and the coefficient of y is 10.
Step 3: Eliminate One of the Variables
Since the coefficients of x in both equations are the same, we can eliminate x by subtracting the first equation from the second equation. This will give us an equation with only one variable, y.
{ \begin{array}{l} -7x + 6y = -13 \\ -7x + 10y = 11 \end{array} \}
Subtracting the first equation from the second equation, we get:
{ (10y - 6y) = (11 - (-13)) \}
Simplifying the equation, we get:
{ 4y = 24 \}
Step 4: Solve for the Variable
Now that we have an equation with only one variable, y, we can solve for y by dividing both sides of the equation by 4.
{ y = \frac{24}{4} \}
Simplifying the equation, we get:
{ y = 6 \}
Step 5: Substitute the Value of the Variable
Now that we have the value of y, we can substitute it into one of the original equations to solve for x. Let's use the first equation:
{ -7x + 6y = -13 \}
Substituting y = 6 into the equation, we get:
{ -7x + 6(6) = -13 \}
Simplifying the equation, we get:
{ -7x + 36 = -13 \}
Subtracting 36 from both sides of the equation, we get:
{ -7x = -49 \}
Dividing both sides of the equation by -7, we get:
{ x = \frac{-49}{-7} \}
Simplifying the equation, we get:
{ x = 7 \}
Conclusion
In this article, we discussed how to solve a system of linear equations using the elimination method. We started by writing down the given equations and identifying the coefficients of x and y. We then eliminated one of the variables by subtracting the first equation from the second equation. We solved for the variable y and then substituted its value into one of the original equations to solve for x. The final solution is x = 7 and y = 6.
Example Problems
Here are a few example problems that you can try to practice the elimination method:
- Solve the system of linear equations using the elimination method:
{ \begin{array}{l} 2x + 3y = 7 \\ 4x + 6y = 12 \end{array} \}
- Solve the system of linear equations using the elimination method:
{ \begin{array}{l} x - 2y = -3 \\ 2x + 4y = 5 \end{array} \}
- Solve the system of linear equations using the elimination method:
{ \begin{array}{l} 3x + 2y = 9 \\ 6x + 4y = 18 \end{array} \}
Tips and Tricks
Here are a few tips and tricks that you can use to make the elimination method easier:
- Make sure to identify the coefficients of x and y in both equations.
- Eliminate one of the variables by subtracting the first equation from the second equation.
- 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.
- Check your solution by plugging it back into both original equations.
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.
Q: When can I use the elimination method?
A: You can use the elimination method when the coefficients of the variables in the equations are the same, but the constants are different.
Q: How do I know which variable to eliminate?
A: You can eliminate either variable, but it's usually easier to eliminate the variable that has the larger coefficient.
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 one or both of the equations by a constant to make the coefficients the same.
Q: How do I eliminate a variable?
A: To eliminate a variable, you can subtract one equation from the other equation. This will give you an equation with only one variable.
Q: What if I get a fraction when I eliminate a variable?
A: If you get a fraction when you eliminate a variable, you can multiply both sides of the equation by the denominator to get rid of the fraction.
Q: How do I solve for the variable?
A: To solve for the variable, you can divide both sides of the equation by the coefficient of the variable.
Q: What if I get a negative value for the variable?
A: If you get a negative value for the variable, you can simply write the negative value as the solution.
Q: Can I use the elimination method with more than two equations?
A: Yes, you can use the elimination method with more than two equations. However, it may be more complicated and require more steps.
Q: What if I get a system of linear equations with no solution?
A: If you get 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 both equations.
Q: What if I get a system of linear equations with infinitely many solutions?
A: If you get 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 both equations.
Q: Can I use the elimination method with systems of linear equations with complex numbers?
A: Yes, you can use the elimination method with systems of linear equations with complex numbers. However, you will need to use complex arithmetic and follow the same steps as with real numbers.
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 identifying the coefficients of the variables correctly
- Not eliminating the correct variable
- Not solving for the variable correctly
- Not checking the solution by plugging it back into both original equations
By following these tips and avoiding common mistakes, you can use the elimination method effectively and solve systems of linear equations with ease.
Example Problems
Here are a few example problems that you can try to practice the elimination method:
- Solve the system of linear equations using the elimination method:
{ \begin{array}{l} 2x + 3y = 7 \\ 4x + 6y = 12 \end{array} \}
- Solve the system of linear equations using the elimination method:
{ \begin{array}{l} x - 2y = -3 \\ 2x + 4y = 5 \end{array} \}
- Solve the system of linear equations using the elimination method:
{ \begin{array}{l} 3x + 2y = 9 \\ 6x + 4y = 18 \end{array} \}
Tips and Tricks
Here are a few tips and tricks that you can use to make the elimination method easier:
- Make sure to identify the coefficients of the variables in both equations.
- Eliminate one of the variables by subtracting the first equation from the second equation.
- 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.
- Check your solution by plugging it back into both original equations.
By following these tips and tricks, you can make the elimination method easier and more efficient.