Solve Each System By Elimination:${ \begin{array}{l} 2x + 6y = -14 \ 2x - 3y = -14 \end{array} }$Write Your { X$}$ Answer As Just The Number In The First Blank. In The Second Blank, Write Just Your { Y$}$ Answer. In The

Mar 9, 2025 by ADMIN 221 views

**Solving Systems of Linear Equations by Elimination**

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and one of the most effective methods for solving these systems is the elimination method. This method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. In this article, we will explore how to solve a system of linear equations using the elimination method.

The Elimination Method

The elimination method involves adding or subtracting equations to eliminate one of the variables. To do this, we need to have two equations with the same coefficient for one of the variables. We can then add or subtract the equations to eliminate that variable.

Example 1: Solving a System of Linear Equations by Elimination

Let's consider the following system of linear equations:

{ \begin{array}{l} 2x + 6y = -14 \\ 2x - 3y = -14 \end{array} \}

To solve this system, we can use the elimination method. First, we need to have the same coefficient for one of the variables. In this case, we can multiply the second equation by 3 to get:

{ \begin{array}{l} 2x + 6y = -14 \\ 6x - 9y = -42 \end{array} \}

Now, we can add the two equations to eliminate the variable x:

{ (2x + 6y) + (6x - 9y) = -14 + (-42) \}

Simplifying the equation, we get:

{ 8x - 3y = -56 \}

Now, we can solve for y by multiplying the first equation by 3 and subtracting the second equation:

{ (6x + 18y) - (8x - 3y) = -42 - (-56) \}

Simplifying the equation, we get:

{ 26y = 14 \}

Dividing both sides by 26, we get:

{ y = \frac{14}{26} = \frac{7}{13} \}

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:

{ 2x + 6y = -14 \}

Substituting y = 7/13, we get:

{ 2x + 6(\frac{7}{13}) = -14 \}

Simplifying the equation, we get:

{ 2x + \frac{42}{13} = -14 \}

Multiplying both sides by 13, we get:

{ 26x + 42 = -182 \}

Subtracting 42 from both sides, we get:

{ 26x = -224 \}

Dividing both sides by 26, we get:

{ x = \frac{-224}{26} = \frac{-112}{13} \}

Therefore, the solution to the system of linear equations is x = -112/13 and y = 7/13.

Discussion

The elimination method is a powerful tool for solving systems of linear equations. By adding or subtracting equations, we can eliminate one of the variables and solve for the other variable. This method is particularly useful when we have two equations with the same coefficient for one of the variables.

In this example, we used the elimination method to solve a system of linear equations. We first multiplied the second equation by 3 to get the same coefficient for x. We then added the two equations to eliminate the variable x. We solved for y by multiplying the first equation by 3 and subtracting the second equation. Finally, we substituted the value of y into one of the original equations to solve for x.

The elimination method is a fundamental concept in mathematics, and it has many applications in real-world problems. By mastering this method, we can solve a wide range of problems involving systems of linear equations.

Conclusion

In conclusion, the elimination method is a powerful tool for solving systems of linear equations. By adding or subtracting equations, we can eliminate one of the variables and solve for the other variable. This method is particularly useful when we have two equations with the same coefficient for one of the variables. By mastering this method, we can solve a wide range of problems involving systems of linear equations.

Example 2: Solving a System of Linear Equations by Elimination

Let's consider the following system of linear equations:

{ \begin{array}{l} x + 2y = 6 \\ 3x - 2y = 2 \end{array} \}

To solve this system, we can use the elimination method. First, we need to have the same coefficient for one of the variables. In this case, we can multiply the first equation by 2 to get:

{ \begin{array}{l} 2x + 4y = 12 \\ 3x - 2y = 2 \end{array} \}

Now, we can add the two equations to eliminate the variable x:

{ (2x + 4y) + (3x - 2y) = 12 + 2 \}

Simplifying the equation, we get:

{ 5x + 2y = 14 \}

Now, we can solve for y by multiplying the second equation by 2 and adding the two equations:

{ (6x - 4y) + (5x + 2y) = 4 + 14 \}

Simplifying the equation, we get:

{ 11x - 2y = 18 \}

Now, we can solve for x by multiplying the first equation by 2 and adding the two equations:

{ (2x + 4y) + (5x + 2y) = 12 + 14 \}

Simplifying the equation, we get:

{ 7x + 6y = 26 \}

Now, we can solve for x by multiplying the second equation by 6 and adding the two equations:

{ (33x - 12y) + (7x + 6y) = 108 + 26 \}

Simplifying the equation, we get:

{ 40x - 6y = 134 \}

Now, we can solve for x by multiplying the first equation by 6 and adding the two equations:

{ (12x + 24y) + (40x - 6y) = 72 + 134 \}

Simplifying the equation, we get:

{ 52x + 18y = 206 \}

Now, we can solve for x by multiplying the second equation by 18 and adding the two equations:

{ (66x - 36y) + (52x + 18y) = 198 + 206 \}

Simplifying the equation, we get:

{ 118x - 18y = 404 \}

Now, we can solve for x by multiplying the first equation by 18 and adding the two equations:

{ (36x + 72y) + (118x - 18y) = 216 + 404 \}

Simplifying the equation, we get:

{ 154x + 54y = 620 \}

Now, we can solve for x by multiplying the second equation by 54 and adding the two equations:

{ (198x - 108y) + (154x + 54y) = 1072 + 620 \}

Simplifying the equation, we get:

{ 352x - 54y = 1692 \}

Now, we can solve for x by multiplying the first equation by 54 and adding the two equations:

{ (1944x + 1944y) + (352x - 54y) = 10416 + 1692 \}

Simplifying the equation, we get:

{ 2296x + 1890y = 12108 \}

Now, we can solve for x by multiplying the second equation by 1890 and adding the two equations:

{ (4338x - 1890y) + (2296x + 1890y) = 8208 + 12108 \}

Simplifying the equation, we get:

{ 6634x = 18416 \}

Dividing both sides by 6634, we get:

{ x = \frac{18416}{6634} = \frac{9208}{3317} \}

Now that we have the value of x, we can substitute it into one of the original equations to solve for y. Let's use the first equation:

{ x + 2y = 6 \}

Substituting x = 9208/3317, we get:

{ \frac{9208}{3317} + 2y = 6 \}

Multiplying both sides by 3317, we get:

{ 9208 + 6624y = 19902 \}

Subtracting 9208 from both sides, we get:

{ 6624y = 10694 \}

Dividing both sides by 6624, we get:

{ y = \frac{10694}{6624} = \frac{5337}{3312} \}

**Solving Systems of Linear Equations by Elimination: Q&A** =====================================================

Introduction

In our previous article, we explored how to solve systems of linear equations using the elimination method. This method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. 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 know which variable to eliminate?

A: To determine which variable to eliminate, you need to have two equations with the same coefficient for one of the variables. You can then add or subtract the equations to eliminate that variable.

Q: What if I have two equations with different coefficients for the same variable?

A: If you have two equations with different coefficients for the same variable, you can multiply one of the equations by a scalar 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 to solve for the third variable.

Q: How do I know if the system of linear equations has a solution?

A: To determine if the system of linear equations has a solution, you can use the following steps:

Add or subtract the equations to eliminate one of the variables.
Solve for the other variable.
Substitute the value of the other variable into one of the original equations to solve for the first variable.
Check if the values of the variables satisfy both equations.

Q: What if the system of linear equations has no solution?

A: If the system of linear equations has no solution, it means that the equations are inconsistent. This can happen if the equations are contradictory, such as 2x + 3y = 5 and 2x + 3y = 7.

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 both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: How do I know if the system of linear equations has a unique solution?

A: To determine if the system of linear equations has a unique solution, you can use the following steps:

Add or subtract the equations to eliminate one of the variables.
Solve for the other variable.
Substitute the value of the other variable into one of the original equations to solve for the first variable.
Check if the values of the variables satisfy both equations.
Check if the values of the variables are unique.

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 multiply both sides of the equation by a power of 10 to eliminate the decimals.

Conclusion

Example 3: Solving a System of Linear Equations by Elimination

Let's consider the following system of linear equations:

{ \begin{array}{l} 2x + 3y = 7 \\ 4x - 2y = -3 \end{array} \}

To solve this system, we can use the elimination method. First, we need to have the same coefficient for one of the variables. In this case, we can multiply the first equation by 2 to get:

{ \begin{array}{l} 4x + 6y = 14 \\ 4x - 2y = -3 \end{array} \}

Now, we can add the two equations to eliminate the variable x:

{ (4x + 6y) + (4x - 2y) = 14 + (-3) \}

Simplifying the equation, we get:

{ 8x + 4y = 11 \}

Now, we can solve for y by multiplying the second equation by 2 and adding the two equations:

{ (8x - 4y) + (8x + 4y) = -6 + 11 \}

Simplifying the equation, we get:

{ 16x = 5 \}

Dividing both sides by 16, we get:

{ x = \frac{5}{16} \}

Now that we have the value of x, we can substitute it into one of the original equations to solve for y. Let's use the first equation:

{ 2x + 3y = 7 \}

Substituting x = 5/16, we get:

{ 2(\frac{5}{16}) + 3y = 7 \}

Simplifying the equation, we get:

{ \frac{5}{8} + 3y = 7 \}

Multiplying both sides by 8, we get:

{ 5 + 24y = 56 \}

Subtracting 5 from both sides, we get:

{ 24y = 51 \}

Dividing both sides by 24, we get:

{ y = \frac{51}{24} = \frac{17}{8} \}

Therefore, the solution to the system of linear equations is x = 5/16 and y = 17/8.