Solve Each System By Elimination:${ \begin{array}{l} -4x + 10y = -10 \ -4x + Y = -19 \end{array} }$Write Your { X$}$ Answer As Just The Number In The First Blank. In The Second Blank, Write Just Your { Y$}$ Answer. In
Introduction
Solving systems of linear equations is a fundamental concept in mathematics, and one of the most effective methods for solving these systems is by elimination. In this article, we will explore how to solve a system of linear equations using the elimination method, and we will apply this method to a specific system of equations.
What is the Elimination Method?
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 based on the principle that if two equations are added or subtracted, the resulting equation will have the same coefficients for the variable being eliminated, but the constant terms will be different.
Step-by-Step Guide to Solving a System of Linear Equations by Elimination
To solve a system of linear equations by elimination, follow these steps:
- Write down the system of equations: Write down the system of linear equations that you want to solve.
- Identify the coefficients and constant terms: Identify the coefficients of the variables and the constant terms in each equation.
- Choose a variable to eliminate: Choose a variable to eliminate, and identify the equations that contain this variable.
- Multiply the equations by necessary multiples: Multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same in both equations.
- Add or subtract the equations: Add or subtract the equations to eliminate the variable.
- Solve for the remaining variable: Solve for the remaining variable.
- Back-substitute to find the other variable: Back-substitute the value of the remaining variable into one of the original equations to find the value of the other variable.
Example: Solving a System of Linear Equations by Elimination
Let's consider the following system of linear equations:
{ \begin{array}{l} -4x + 10y = -10 \\ -4x + y = -19 \end{array} \}
To solve this system by elimination, we will follow the steps outlined above.
Step 1: Write down the system of equations
The system of equations is:
{ \begin{array}{l} -4x + 10y = -10 \\ -4x + y = -19 \end{array} \}
Step 2: Identify the coefficients and constant terms
The coefficients and constant terms are:
| Equation | Coefficients | Constant Terms |
|---|---|---|
| 1 | -4, 10 | -10 |
| 2 | -4, 1 | -19 |
Step 3: Choose a variable to eliminate
We will choose to eliminate the variable x.
Step 4: Multiply the equations by necessary multiples
We will multiply the first equation by 1 and the second equation by 10, such that the coefficients of x are the same in both equations.
{ \begin{array}{l} -4x + 10y = -10 \\ 40x + 10y = -190 \end{array} \}
Step 5: Add or subtract the equations
We will add the two equations to eliminate the variable x.
{ \begin{array}{l} -4x + 10y = -10 \\ 40x + 10y = -190 \end{array} \}
{ \begin{array}{l} 36x + 20y = -180 \end{array} \}
Step 6: Solve for the remaining variable
We will solve for the remaining variable y.
{ \begin{array}{l} 36x + 20y = -180 \end{array} \}
{ \begin{array}{l} y = \frac{-180 - 36x}{20} \end{array} \}
Step 7: Back-substitute to find the other variable
We will back-substitute the value of y into one of the original equations to find the value of x.
{ \begin{array}{l} -4x + 10y = -10 \end{array} \}
{ \begin{array}{l} -4x + 10(\frac{-180 - 36x}{20}) = -10 \end{array} \}
{ \begin{array}{l} -4x + (-90 - 18x) = -10 \end{array} \}
{ \begin{array}{l} -22x = 80 \end{array} \}
{ \begin{array}{l} x = \frac{-80}{22} \end{array} \}
{ \begin{array}{l} x = -\frac{40}{11} \end{array} \}
Step 8: Find the value of the other variable
We will substitute the value of x into one of the original equations to find the value of y.
{ \begin{array}{l} -4x + y = -19 \end{array} \}
{ \begin{array}{l} -4(-\frac{40}{11}) + y = -19 \end{array} \}
{ \begin{array}{l} \frac{160}{11} + y = -19 \end{array} \}
{ \begin{array}{l} y = -19 - \frac{160}{11} \end{array} \}
{ \begin{array}{l} y = \frac{-209 - 160}{11} \end{array} \}
{ \begin{array}{l} y = \frac{-369}{11} \end{array} \}
{ \begin{array}{l} y = -\frac{369}{11} \end{array} \}
Conclusion
In this article, we have explored how to solve a system of linear equations using the elimination method. We have applied this method to a specific system of equations and have found the values of the variables x and y. The elimination method is a powerful tool for solving systems of linear equations, and it is an essential concept in mathematics.
Final Answer
The final answer is:
Introduction
In our previous article, we explored how to solve a system of linear equations using the elimination method. In this article, we will answer some frequently asked questions about solving systems of linear equations by elimination.
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: How do I choose a variable to eliminate?
A: To choose a variable to eliminate, look for the equations that contain the variable you want to eliminate. Then, multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same in both equations.
Q: What if the coefficients of the variable to be eliminated are not the same in both equations?
A: If the coefficients of the variable to be eliminated are not the same in both equations, you can multiply one or both of the equations by a constant to make the coefficients the same.
Q: How do I add or subtract the equations to eliminate the variable?
A: To add or subtract the equations, add or subtract the coefficients of the variable to be eliminated and the constant terms.
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 fractions or decimals.
Q: How do I solve for the remaining variable?
A: To solve for the remaining variable, isolate the variable on one side of the equation by adding or subtracting the same value to both sides of the equation.
Q: What if I get a quadratic equation when I solve for the remaining variable?
A: If you get a quadratic equation when you solve for the remaining variable, you can use the quadratic formula to solve for the variable.
Q: How do I back-substitute to find the other variable?
A: To back-substitute, substitute the value of the remaining variable into one of the original equations to find the value of the other variable.
Q: What if I get a fraction or a decimal when I back-substitute?
A: If you get a fraction or a decimal when you back-substitute, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions or decimals.
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 to solve for the third variable.
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 the variable to be eliminated the same.
- Not adding or subtracting the equations correctly.
- Not solving for the remaining variable correctly.
- Not back-substituting correctly.
Conclusion
In this article, we have answered some frequently asked questions about solving systems of linear equations by elimination. We hope that this article has been helpful in clarifying any confusion you may have had about the elimination method.
Final Tips
- Make sure to read the problem carefully and understand what is being asked.
- Choose a variable to eliminate that is easy to work with.
- Multiply the equations by necessary multiples to make the coefficients of the variable to be eliminated the same.
- Add or subtract the equations correctly.
- Solve for the remaining variable correctly.
- Back-substitute correctly.
By following these tips and avoiding common mistakes, you can use the elimination method to solve systems of linear equations with ease.