Use The Elimination Method To Solve The System Of Equations. Choose The Correct Ordered Pair.${ \begin{array}{l} 7x + 4y = 39 \ 2x + 4y = 14 \end{array} }$A. { (3,4)$}$ B. { (5,1)$}$ C. { (5,4)$}$ D.

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Introduction

Solving systems of 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 use the elimination method to solve a system of two linear equations and choose the correct ordered pair.

The Elimination Method

The elimination method is a step-by-step process that involves the following steps:

  1. Write down the system of equations: Write down the two linear equations that make up the system.
  2. Multiply the equations by necessary multiples: Multiply one or both of the equations by necessary multiples such that the coefficients of one of the variables (either x or y) are the same in both equations.
  3. Add or subtract the equations: Add or subtract the equations to eliminate one of the variables.
  4. Solve for the other variable: Solve for the other variable using the resulting equation.
  5. Check the solution: Check the solution by plugging it back into both original equations.

Solving the System of Equations

Let's use the elimination method to solve the system of equations:

{ \begin{array}{l} 7x + 4y = 39 \\ 2x + 4y = 14 \end{array} \}

Step 1: Write down the system of equations

We have already written down the system of equations:

{ \begin{array}{l} 7x + 4y = 39 \\ 2x + 4y = 14 \end{array} \}

Step 2: Multiply the equations by necessary multiples

We need to multiply the first equation by 1 and the second equation by 7, so that the coefficients of x are the same in both equations.

{ \begin{array}{l} 7x + 4y = 39 \\ 14x + 28y = 98 \end{array} \}

Step 3: Add or subtract the equations

We can subtract the first equation from the second equation to eliminate the variable x.

(14xβˆ’7x)+(28yβˆ’4y)=98βˆ’39{ (14x - 7x) + (28y - 4y) = 98 - 39 }

Simplifying the equation, we get:

7x+24y=59{ 7x + 24y = 59 }

Step 4: Solve for the other variable

We can solve for y by subtracting 7x from both sides of the equation.

24y=59βˆ’7x{ 24y = 59 - 7x }

Dividing both sides by 24, we get:

y=59βˆ’7x24{ y = \frac{59 - 7x}{24} }

Step 5: Check the solution

We can plug the value of y back into one of the original equations to check the solution.

Let's plug y = (59 - 7x)/24 into the first equation:

7x+4(59βˆ’7x24)=39{ 7x + 4\left(\frac{59 - 7x}{24}\right) = 39 }

Simplifying the equation, we get:

7x+236βˆ’28x6=39{ 7x + \frac{236 - 28x}{6} = 39 }

Multiplying both sides by 6, we get:

42x+236βˆ’28x=234{ 42x + 236 - 28x = 234 }

Simplifying the equation, we get:

14x=βˆ’2{ 14x = -2 }

Dividing both sides by 14, we get:

x=βˆ’214=βˆ’17{ x = -\frac{2}{14} = -\frac{1}{7} }

Plugging x = -1/7 back into the equation y = (59 - 7x)/24, we get:

y=59βˆ’7(βˆ’17)24=59+124=6024=52{ y = \frac{59 - 7\left(-\frac{1}{7}\right)}{24} = \frac{59 + 1}{24} = \frac{60}{24} = \frac{5}{2} }

So, the solution to the system of equations is x = -1/7 and y = 5/2.

Choosing the Correct Ordered Pair

Now that we have solved the system of equations, we need to choose the correct ordered pair from the options given.

A. (3,4) B. (5,1) C. (5,4) D. (other options)

We can plug each of the ordered pairs into one of the original equations to check if it is a solution.

Let's plug (3,4) into the first equation:

7(3)+4(4)=21+16=37β‰ 39{ 7(3) + 4(4) = 21 + 16 = 37 \neq 39 }

So, (3,4) is not a solution.

Let's plug (5,1) into the first equation:

7(5)+4(1)=35+4=39{ 7(5) + 4(1) = 35 + 4 = 39 }

So, (5,1) is a solution.

Let's plug (5,4) into the first equation:

7(5)+4(4)=35+16=51β‰ 39{ 7(5) + 4(4) = 35 + 16 = 51 \neq 39 }

So, (5,4) is not a solution.

Therefore, the correct ordered pair is (5,1).

Conclusion

In this article, we used the elimination method to solve a system of two linear equations and chose the correct ordered pair. We wrote down the system of equations, multiplied the equations by necessary multiples, added or subtracted the equations, solved for the other variable, and checked the solution. We then plugged each of the ordered pairs into one of the original equations to check if it was a solution. Finally, we chose the correct ordered pair from the options given.

Introduction

Solving systems of equations is a fundamental concept in mathematics, and one of the most effective methods for solving these systems is the elimination method. In this article, we will answer some frequently asked questions (FAQs) about solving systems of equations using the elimination method.

Q: What is the elimination method?

A: The elimination method is a step-by-step process that 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?

A: You can eliminate either variable, but it's often easier to eliminate the variable that has the smaller coefficient.

Q: What if the coefficients of the variables are the same?

A: If the coefficients of the variables are the same, you can multiply one or both of the equations by a constant to make the coefficients different.

Q: Can I use the elimination method with systems of equations that have more than two variables?

A: Yes, you can use the elimination method with systems of equations that have more than two variables. However, it may be more complicated and require more steps.

Q: What if I get a false solution?

A: If you get a false solution, it means that the system of equations has no solution. This can happen if the equations are inconsistent.

Q: Can I use the elimination method with systems of equations that have fractions?

A: Yes, you can use the elimination method with systems of equations that have fractions. However, you may need to multiply the equations by a constant to eliminate the fractions.

Q: What if I get a solution that doesn't satisfy one of the original equations?

A: If you get a solution that doesn't satisfy one of the original equations, it means that the solution is not valid. You should go back and check your work to see where you made a mistake.

Q: Can I use the elimination method with systems of equations that have decimals?

A: Yes, you can use the elimination method with systems of equations that have decimals. However, you may need to multiply the equations by a constant to eliminate the decimals.

Q: What if I'm not sure which method to use?

A: If you're not sure which method to use, you can try using the substitution method or the graphing method to see if you get the same solution.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about solving systems of equations using the elimination method. We covered topics such as how to know which variable to eliminate, what to do if the coefficients of the variables are the same, and how to handle systems of equations with fractions, decimals, and more than two variables. We hope that this article has been helpful in answering your questions and providing you with a better understanding of the elimination method.

Additional Resources

If you're looking for more information on solving systems of equations using the elimination method, here are some additional resources that you may find helpful:

  • Online tutorials: There are many online tutorials and videos that can help you learn how to solve systems of equations using the elimination method.
  • Practice problems: You can find practice problems and worksheets online that can help you practice solving systems of equations using the elimination method.
  • Math textbooks: If you're looking for a more in-depth explanation of the elimination method, you can find it in math textbooks that cover algebra and systems of equations.
  • Math software: There are many math software programs that can help you solve systems of equations using the elimination method, such as Mathematica and Maple.

We hope that this article has been helpful in answering your questions and providing you with a better understanding of the elimination method. If you have any further questions or need additional help, don't hesitate to ask.