Solve The Following Systems Of Equations By Elimination:1.${ \begin{align*} x + 3y &= 8 \ 5x + 7y &= 24 \end{align*} } 2. 2. 2. ${ \begin{align*} x + Y &= 3 \ 3x - Y &= 5 \end{align*} } 3. 3. 3. [ \begin{align*} 5x + 4y &= 22 \ 4x + 5y

Introduction

Systems of equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving systems of 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.

What is the Elimination Method?

The elimination method is a technique used to solve systems of equations by adding or subtracting equations to eliminate one of the variables. This method is useful when the coefficients of the variables in the equations are not the same. By adding or subtracting the equations, we can create a new equation with one variable eliminated, making it easier to solve for the other variable.

Step-by-Step Guide to Solving Systems of Equations by Elimination

To solve a system of equations using the elimination method, follow these steps:

1. Write down the equations: Write down the two equations in the system.
2. Identify the coefficients: Identify the coefficients of the variables in both equations.
3. Determine the elimination method: Determine which variable to eliminate by comparing the coefficients of the variables in both equations.
4. Add or subtract the equations: Add or subtract the equations to eliminate one of the variables.
5. Solve for the other variable: Solve for the other variable using the new equation.
6. Check the solution: Check the solution by plugging it back into both original equations.

Example 1: Solving a System of Equations by Elimination

Let's solve the following system of equations using the elimination method:

{ \begin{align*} x + 3y &= 8 \\ 5x + 7y &= 24 \end{align*} \}

Step 1: Write down the equations

The two equations in the system are:

$x + 3y = 8$

$5x + 7y = 24$

Step 2: Identify the coefficients

The coefficients of the variables in both equations are:

$x$: 1, 5

$y$: 3, 7

Step 3: Determine the elimination method

To eliminate one of the variables, we need to compare the coefficients of the variables in both equations. In this case, we can eliminate the variable $x$ by subtracting the first equation from the second equation.

Step 4: Add or subtract the equations

Subtract the first equation from the second equation:

$(5x + 7y) - (x + 3y) = 24 - 8$

Simplify the equation:

$4x + 4y = 16$

Step 5: Solve for the other variable

Solve for the variable $x$ using the new equation:

$4x + 4y = 16$

Subtract $4y$ from both sides:

$4x = 16 - 4y$

Divide both sides by 4:

$x = 4 - y$

Step 6: Check the solution

Check the solution by plugging it back into both original equations:

$x + 3y = 8$

Substitute $x = 4 - y$ into the equation:

$(4 - y) + 3y = 8$

Simplify the equation:

$4 + 2y = 8$

Subtract 4 from both sides:

$2y = 4$

Divide both sides by 2:

$y = 2$

Substitute $y = 2$ into the equation $x = 4 - y$:

$x = 4 - 2$

Simplify the equation:

$x = 2$

Conclusion

The solution to the system of equations is $x = 2$ and $y = 2$.

Example 2: Solving a System of Equations by Elimination

Let's solve the following system of equations using the elimination method:

{ \begin{align*} x + y &= 3 \\ 3x - y &= 5 \end{align*} \}

Step 1: Write down the equations

The two equations in the system are:

$x + y = 3$

$3x - y = 5$

Step 2: Identify the coefficients

The coefficients of the variables in both equations are:

$x$: 1, 3

$y$: 1, -1

Step 3: Determine the elimination method

To eliminate one of the variables, we need to compare the coefficients of the variables in both equations. In this case, we can eliminate the variable $y$ by adding the two equations.

Step 4: Add or subtract the equations

Add the two equations:

$(x + y) + (3x - y) = 3 + 5$

Simplify the equation:

$4x = 8$

Step 5: Solve for the other variable

Solve for the variable $x$ using the new equation:

$4x = 8$

Divide both sides by 4:

$x = 2$

Step 6: Check the solution

Check the solution by plugging it back into both original equations:

$x + y = 3$

Substitute $x = 2$ into the equation:

$2 + y = 3$

Subtract 2 from both sides:

$y = 1$

Conclusion

The solution to the system of equations is $x = 2$ and $y = 1$.

Example 3: Solving a System of Equations by Elimination

Let's solve the following system of equations using the elimination method:

{ \begin{align*} 5x + 4y &= 22 \\ 4x + 5y &= 31 \end{align*} \}

Step 1: Write down the equations

The two equations in the system are:

$5x + 4y = 22$

$4x + 5y = 31$

Step 2: Identify the coefficients

The coefficients of the variables in both equations are:

$x$: 5, 4

$y$: 4, 5

Step 3: Determine the elimination method

To eliminate one of the variables, we need to compare the coefficients of the variables in both equations. In this case, we can eliminate the variable $x$ by multiplying the first equation by 4 and the second equation by 5, and then subtracting the two equations.

Step 4: Add or subtract the equations

Multiply the first equation by 4:

$20x + 16y = 88$

Multiply the second equation by 5:

$20x + 25y = 155$

Subtract the two equations:

$(20x + 25y) - (20x + 16y) = 155 - 88$

Simplify the equation:

$9y = 67$

Step 5: Solve for the other variable

Solve for the variable $y$ using the new equation:

$9y = 67$

Divide both sides by 9:

$y = \frac{67}{9}$

Step 6: Check the solution

Check the solution by plugging it back into both original equations:

$5x + 4y = 22$

Substitute $y = \frac{67}{9}$ into the equation:

$5x + 4(\frac{67}{9}) = 22$

Simplify the equation:

$5x + \frac{268}{9} = 22$

Multiply both sides by 9:

$45x + 268 = 198$

Subtract 268 from both sides:

$45x = -70$

Divide both sides by 45:

$x = -\frac{70}{45}$

Conclusion

The solution to the system of equations is $x = -\frac{70}{45}$ and $y = \frac{67}{9}$.

Conclusion

Q: What is the elimination method for solving systems of equations?

A: The elimination method is a technique used to solve systems of equations by adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable.

Q: When should I use the elimination method to solve a system of equations?

A: You should use the elimination method when the coefficients of the variables in the equations are not the same. This method is useful when you want to eliminate one of the variables by adding or subtracting the equations.

Q: How do I determine which variable to eliminate when using the elimination method?

A: To determine which variable to eliminate, compare the coefficients of the variables in both equations. You can eliminate the variable with the smaller coefficient by adding or subtracting the equations.

Q: What are the steps involved in solving a system of equations using the elimination method?

A: The steps involved in solving a system of equations using the elimination method are:

1. Write down the equations
2. Identify the coefficients
3. Determine the elimination method
4. Add or subtract the equations
5. Solve for the other variable
6. Check the solution

Q: Can I use the elimination method to solve a system of equations with three variables?

A: No, the elimination method is used to solve systems of equations with two variables. If you have a system of equations with three variables, you will need to use a different method, such as substitution or graphing.

Q: What if I get a fraction or decimal when solving a system of equations using the elimination method?

A: If you get a fraction or decimal when solving a system of equations using the elimination method, you can simplify the solution by dividing both sides of the equation by the denominator or multiplying both sides by a power of 10.

Q: Can I use the elimination method to solve a system of equations with negative coefficients?

A: Yes, you can use the elimination method to solve a system of equations with negative coefficients. The steps involved in solving a system of equations with negative coefficients are the same as those for solving a system of equations with positive coefficients.

Q: What if I get a system of equations with no solution using the elimination method?

A: If you get a system of equations with no solution using the elimination method, it means that the equations are inconsistent and there is no solution to the system.

Q: Can I use the elimination method to solve a system of equations with infinitely many solutions?

A: Yes, you can use the elimination method to solve a system of equations with infinitely many solutions. If the equations are dependent, the elimination method will result in an equation with no solution, indicating that the system has infinitely many solutions.

Q: What are some common mistakes to avoid when using the elimination method to solve a system of equations?

A: Some common mistakes to avoid when using the elimination method to solve a system of equations include:

• Not following the steps involved in the elimination method
• Not checking the solution by plugging it back into both original equations
• Not simplifying the solution by dividing both sides of the equation by the denominator or multiplying both sides by a power of 10
• Not recognizing that the equations are inconsistent or dependent

Conclusion

In this article, we have answered some frequently asked questions about solving systems of equations by elimination. We have discussed the steps involved in the elimination method, how to determine which variable to eliminate, and how to check the solution. We have also discussed some common mistakes to avoid when using the elimination method to solve a system of equations. By following the steps outlined in this article, you can solve systems of equations using the elimination method.