Solve The System Of Equations By Elimination:$\[ \begin{cases} -4x - 6y = 4 \\ -3x - 6y = 0 \end{cases} \\]Solution: \[$\square\$\], \[$\square\$\]
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Introduction
In this article, we will learn how to solve a system of linear equations using the elimination method. The elimination method is a popular technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. We will use a system of two linear equations in two variables to demonstrate the elimination method.
What is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation in the system is a linear equation, which means that it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are variables.
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. The basic idea behind the elimination method is to add or subtract equations in such a way that one of the variables is eliminated.
Step 1: Write Down the System of Equations
The first step in solving a system of linear equations using the elimination method is to write down the system of equations. In this case, we have the following system of equations:
{ \begin{cases} -4x - 6y = 4 \\ -3x - 6y = 0 \end{cases} \}
Step 2: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to multiply the equations by necessary multiples. In this case, we can multiply the first equation by 3 and the second equation by 4 to make the coefficients of x in both equations equal.
{ \begin{cases} -12x - 18y = 12 \\ -12x - 24y = 0 \end{cases} \}
Step 3: Subtract the Second Equation from the First Equation
Now that we have the coefficients of x in both equations equal, we can subtract the second equation from the first equation to eliminate the variable x.
{ -18y + 24y = 12 - 0 \}
Simplifying the equation, we get:
{ 6y = 12 \}
Step 4: Solve for y
Now that we have the equation 6y = 12, we can solve for y by dividing both sides of the equation by 6.
{ y = \frac{12}{6} \}
Simplifying the equation, we get:
{ y = 2 \}
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 y = 2 into the first equation.
{ -4x - 6(2) = 4 \}
Simplifying the equation, we get:
{ -4x - 12 = 4 \}
Adding 12 to both sides of the equation, we get:
{ -4x = 16 \}
Dividing both sides of the equation by -4, we get:
{ x = -4 \}
Conclusion
In this article, we learned how to solve a system of linear equations using the elimination method. We used a system of two linear equations in two variables to demonstrate the elimination method. We 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.
Discussion
The elimination method is a popular technique used to solve systems of linear equations. It is a simple and effective method that can be used to solve systems of linear equations with two or more variables. The elimination method involves adding or subtracting equations to eliminate one of the variables, and then solving for the other variable.
Advantages of the Elimination Method
The elimination method has several advantages. It is a simple and easy-to-use method that can be used to solve systems of linear equations with two or more variables. It is also a fast and efficient method that can be used to solve systems of linear equations.
Disadvantages of the Elimination Method
The elimination method also has several disadvantages. It can be difficult to use the elimination method to solve systems of linear equations with fractions or decimals. It can also be difficult to use the elimination method to solve systems of linear equations with multiple variables.
Examples
Here are a few examples of systems of linear equations that can be solved using the elimination method.
Example 1
Solve the system of linear equations:
{ \begin{cases} 2x + 3y = 5 \\ x - 2y = -3 \end{cases} \}
Using the elimination method, we can multiply the first equation by 2 and the second equation by 3 to make the coefficients of x in both equations equal.
{ \begin{cases} 4x + 6y = 10 \\ 3x - 6y = -9 \end{cases} \}
Subtracting the second equation from the first equation, we get:
{ 10y = 19 \}
Solving for y, we get:
{ y = \frac{19}{10} \}
Substituting y = 19/10 into the first equation, we get:
{ 2x + 3(\frac{19}{10}) = 5 \}
Simplifying the equation, we get:
{ 2x + \frac{57}{10} = 5 \}
Multiplying both sides of the equation by 10, we get:
{ 20x + 57 = 50 \}
Subtracting 57 from both sides of the equation, we get:
{ 20x = -7 \}
Dividing both sides of the equation by 20, we get:
{ x = -\frac{7}{20} \}
Example 2
Solve the system of linear equations:
{ \begin{cases} x + 2y = 3 \\ 2x + 4y = 6 \end{cases} \}
Using the elimination method, we can multiply the first equation by 2 and the second equation by 1 to make the coefficients of x in both equations equal.
{ \begin{cases} 2x + 4y = 6 \\ 2x + 4y = 6 \end{cases} \}
Subtracting the second equation from the first equation, we get:
{ 0 = 0 \}
This means that the system of linear equations has infinitely many solutions.
Conclusion
In this article, we learned how to solve a system of linear equations using the elimination method. We used a system of two linear equations in two variables to demonstrate the elimination method. We 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 elimination method is a popular technique used to solve systems of linear equations. It is a simple and effective method that can be used to solve systems of linear equations with two or more variables. The elimination method involves adding or subtracting equations to eliminate one of the variables, and then solving for the other variable.
We also discussed the advantages and disadvantages of the elimination method, and provided examples of systems of linear equations that can be solved using the elimination method.
References
- [1] "Linear Algebra and Its Applications" by Gilbert Strang
- [2] "Introduction to Linear Algebra" by Gilbert Strang
- [3] "Linear Algebra: A Modern Introduction" by David Poole
Keywords
- System of linear equations
- Elimination method
- Linear algebra
- Mathematics
- Algebra
- Equations
- Variables
- Coefficients
- Multiples
- Subtraction
- Addition
- Solving for x
- Solving for y
- Infinitely many solutions
- No solution
- Unique solution
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Introduction
In this article, we will answer some frequently asked questions (FAQs) about solving systems of linear equations by elimination. The elimination method is a popular technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.
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. The basic idea behind the elimination method is to add or subtract equations in such a way that one of the variables is eliminated.
Q: How do I know which variable to eliminate?
A: To determine which variable to eliminate, you need to look at the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, you can eliminate that variable by subtracting one equation from the other.
Q: What if the coefficients of the variables are not the same?
A: If the coefficients of the variables are not the same, you need to multiply one or both of the equations by a necessary multiple to make the coefficients of the variables the same. This will allow you to eliminate one of the variables by subtracting one equation from the other.
Q: How do I multiply an equation by a necessary multiple?
A: To multiply an equation by a necessary multiple, you need to multiply each term in the equation by the same multiple. For example, if you want to multiply the equation 2x + 3y = 5 by 2, you would multiply each term in the equation by 2 to get 4x + 6y = 10.
Q: What if I get a zero coefficient for one of the variables?
A: If you get a zero coefficient for one of the variables, it means that the variable is eliminated. You can then solve for the other variable by substituting the value of the eliminated variable into one of the original equations.
Q: What if I get a zero coefficient for both variables?
A: If you get a zero coefficient for both variables, it means that the system of linear equations has infinitely many solutions. This is because the two equations are essentially the same, and there are many possible values for the variables that satisfy both equations.
Q: What if I get a contradiction?
A: If you get a contradiction, it means that the system of linear equations has no solution. This is because the two equations are inconsistent, and there is no value for the variables that satisfies both equations.
Q: Can I use the elimination method to solve systems of linear equations with fractions or decimals?
A: Yes, you can use the elimination method to solve systems of linear equations with fractions or decimals. However, you need to be careful when multiplying equations by necessary multiples, as this can lead to fractions or decimals in the resulting equation.
Q: Can I use the elimination method to solve systems of linear equations with multiple variables?
A: Yes, you can use the elimination method to solve systems of linear equations with multiple variables. However, you need to be careful when multiplying equations by necessary multiples, as this can lead to complex equations with multiple variables.
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 equations by necessary multiples to make the coefficients of the variables the same
- Not subtracting one equation from the other to eliminate one of the variables
- Not solving for the remaining variable after eliminating one of the variables
- Not checking for zero coefficients or contradictions
Conclusion
In this article, we answered some frequently asked questions (FAQs) about solving systems of linear equations by elimination. We discussed the elimination method, how to determine which variable to eliminate, how to multiply equations by necessary multiples, and how to avoid common mistakes. We also discussed the importance of checking for zero coefficients or contradictions when using the elimination method.
References
- [1] "Linear Algebra and Its Applications" by Gilbert Strang
- [2] "Introduction to Linear Algebra" by Gilbert Strang
- [3] "Linear Algebra: A Modern Introduction" by David Poole
Keywords
- System of linear equations
- Elimination method
- Linear algebra
- Mathematics
- Algebra
- Equations
- Variables
- Coefficients
- Multiples
- Subtraction
- Addition
- Solving for x
- Solving for y
- Infinitely many solutions
- No solution
- Unique solution