Solve The Following Systems Of Equations By Elimination:1. ${ \begin{aligned} x + 3y &= 8 \ 5x + 7y &= 24 \end{aligned} }$2. ${ \begin{aligned} 5x + 4y &= 22 \ 4x + 5y &= 23 \end{aligned} }$3. $[ \begin{aligned} 5x - 3y &=

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 linear equations. It involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. The goal is to eliminate one of the variables by making the coefficients of that variable the same in both equations, but with opposite signs.

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

Step 1: Write Down the Equations

The first step in solving a system of equations by elimination is to write down the equations. Make sure to write the equations in the correct format, with the variables on one side and the constants on the other.

Step 2: Identify the Coefficients

The next step is to identify the coefficients of the variables in both equations. The coefficients are the numbers that multiply the variables.

Step 3: Make the Coefficients of One Variable the Same

The goal is to make the coefficients of one variable the same in both equations, but with opposite signs. This can be done by multiplying one or both of the equations by a constant.

Step 4: Add or Subtract the Equations

Once the coefficients of one variable are the same, add or subtract the equations to eliminate that variable.

Step 5: Solve for the Other Variable

After eliminating one variable, solve for the other variable by dividing both sides of the equation by the coefficient of the other variable.

Step 6: Check the Solution

Finally, check the solution by plugging the values back into both original equations.

Example 1: Solving the System of Equations

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

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

Step 1: Write Down the Equations

The first step is to write down the equations:

$x + 3y = 8$

$5x + 7y = 24$

Step 2: Identify the Coefficients

The next step is to identify the coefficients of the variables:

$x$: 1, 5

$y$: 3, 7

Step 3: Make the Coefficients of One Variable the Same

To make the coefficients of $x$ the same, multiply the first equation by 5 and the second equation by 1:

$5x + 15y = 40$

$5x + 7y = 24$

Step 4: Add or Subtract the Equations

Now, subtract the second equation from the first equation to eliminate $x$:

$(5x + 15y) - (5x + 7y) = 40 - 24$

$8y = 16$

Step 5: Solve for the Other Variable

Divide both sides of the equation by 8 to solve for $y$:

$y = 16/8$

$y = 2$

Step 6: Check the Solution

Plug the value of $y$ back into one of the original equations to check the solution:

$x + 3(2) = 8$

$x + 6 = 8$

$x = 2$

Example 2: Solving the System of Equations

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

{ \begin{aligned} 5x + 4y &= 22 \\ 4x + 5y &= 23 \end{aligned} \}

Step 1: Write Down the Equations

The first step is to write down the equations:

$5x + 4y = 22$

$4x + 5y = 23$

Step 2: Identify the Coefficients

The next step is to identify the coefficients of the variables:

$x$: 5, 4

$y$: 4, 5

Step 3: Make the Coefficients of One Variable the Same

To make the coefficients of $y$ the same, multiply the first equation by 5 and the second equation by 4:

$25x + 20y = 110$

$16x + 20y = 92$

Step 4: Add or Subtract the Equations

Now, subtract the second equation from the first equation to eliminate $y$:

$(25x + 20y) - (16x + 20y) = 110 - 92$

$9x = 18$

Step 5: Solve for the Other Variable

Divide both sides of the equation by 9 to solve for $x$:

$x = 18/9$

$x = 2$

Step 6: Check the Solution

Plug the value of $x$ back into one of the original equations to check the solution:

$5(2) + 4y = 22$

$10 + 4y = 22$

$4y = 12$

$y = 3$

Example 3: Solving the System of Equations

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

{ \begin{aligned} 5x - 3y &= 11 \\ 4x + 5y &= 23 \end{aligned} \}

Step 1: Write Down the Equations

The first step is to write down the equations:

$5x - 3y = 11$

$4x + 5y = 23$

Step 2: Identify the Coefficients

The next step is to identify the coefficients of the variables:

$x$: 5, 4

$y$: -3, 5

Step 3: Make the Coefficients of One Variable the Same

To make the coefficients of $x$ the same, multiply the first equation by 4 and the second equation by 5:

$20x - 12y = 44$

$20x + 25y = 115$

Step 4: Add or Subtract the Equations

Now, subtract the first equation from the second equation to eliminate $x$:

$(20x + 25y) - (20x - 12y) = 115 - 44$

$37y = 71$

Step 5: Solve for the Other Variable

Divide both sides of the equation by 37 to solve for $y$:

$y = 71/37$

$y = 1.946$

Step 6: Check the Solution

Plug the value of $y$ back into one of the original equations to check the solution:

$5x - 3(1.946) = 11$

$5x - 5.838 = 11$

$5x = 16.838$

$x = 3.368$

Conclusion

Q: What is the elimination method?

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

Q: How do I know which variable to eliminate first?

A: To determine which variable to eliminate first, look at the coefficients of the variables in both equations. If the coefficients of one variable are the same, but with opposite signs, eliminate that variable. If not, multiply one or both of the equations by a constant to make the coefficients of one variable the same.

Q: What if I have a system of equations with three variables?

A: If you have a system of equations with three variables, you can use the elimination method to solve for two of the variables, and then use substitution to solve for the third variable.

Q: Can I use the elimination method to solve systems of equations with fractions?

A: Yes, you can use the elimination method to solve systems of equations with fractions. However, you may need to multiply both equations by a common denominator to eliminate the fractions.

Q: How do I check my solution?

A: To check your solution, plug the values of the variables back into both original equations. If the equations are true, then your solution is correct.

Q: What if I get a system of equations with no solution?

A: If you get a system of equations with no solution, it means that the equations are inconsistent. This can happen if the equations are contradictory, or if the system is inconsistent.

Q: Can I use the elimination method to solve systems of equations with decimals?

A: Yes, you can use the elimination method to solve systems of equations with decimals. However, you may need to round the decimals to a certain number of places to make the calculations easier.

Q: How do I know if I should use the elimination method or substitution method?

A: To determine whether to use the elimination method or substitution method, look at the coefficients of the variables in both equations. If the coefficients of one variable are the same, but with opposite signs, use the elimination method. If not, use the substitution method.

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

A: Yes, you can use the elimination method to solve systems of equations with negative coefficients. However, you may need to multiply one or both of the equations by a negative constant to make the coefficients of one variable the same.

Q: How do I handle systems of equations with complex numbers?

A: To handle systems of equations with complex numbers, use the same steps as you would with real numbers. However, be careful when multiplying and dividing complex numbers.

Q: Can I use the elimination method to solve systems of equations with variables on both sides of the equation?

A: No, you cannot use the elimination method to solve systems of equations with variables on both sides of the equation. In this case, you would need to use the substitution method.

Conclusion

Solving systems of equations by elimination is a powerful technique that can be used to solve a wide range of problems. By following the steps outlined in this article, you can easily solve systems of equations using the elimination method. Remember to identify the coefficients of the variables, make the coefficients of one variable the same, add or subtract the equations, solve for the other variable, and check the solution. With practice, you will become proficient in solving systems of equations using the elimination method.