Solve The Following System Of Equations Using The Substitution Method:$\[ \begin{cases} m + N = 5 \\ m - N = 3 \end{cases} \\]Choose The Correct Solution:A. There Is No Solution. B. There Are An Infinite Number Of Solutions. C. The Solution

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Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. There are several methods to solve a system of equations, including the substitution method, the elimination method, and the graphing method. In this article, we will focus on solving a system of equations using the substitution method.

What is the Substitution Method?

The substitution method is a technique used to solve a system of equations by substituting one equation into the other equation. This method is useful when one of the equations can be easily solved for one of the variables. The substitution method involves the following steps:

  1. Solve one of the equations for one of the variables.
  2. Substitute the expression for the variable into the other equation.
  3. Solve the resulting equation for the other variable.
  4. Back-substitute the value of the second variable into one of the original equations to find the value of the first variable.

Solving the System of Equations

Let's consider the following system of equations:

{ \begin{cases} m + n = 5 \\ m - n = 3 \end{cases} \}

To solve this system of equations using the substitution method, we can follow the steps outlined above.

Step 1: Solve one of the equations for one of the variables

We can solve the first equation for m:

m = 5 - n

Step 2: Substitute the expression for the variable into the other equation

Substitute the expression for m into the second equation:

5 - n - n = 3

Step 3: Solve the resulting equation for the other variable

Combine like terms:

5 - 2n = 3

Subtract 5 from both sides:

-2n = -2

Divide both sides by -2:

n = 1

Step 4: Back-substitute the value of the second variable into one of the original equations to find the value of the first variable

Substitute n = 1 into the first equation:

m + 1 = 5

Subtract 1 from both sides:

m = 4

Conclusion

In this article, we solved a system of equations using the substitution method. We first solved one of the equations for one of the variables, then substituted the expression for the variable into the other equation. We solved the resulting equation for the other variable and back-substituted the value of the second variable into one of the original equations to find the value of the first variable. The solution to the system of equations is m = 4 and n = 1.

Answer

The correct solution to the system of equations is:

m = 4 and n = 1

Therefore, the correct answer is:

C. The solution is m = 4 and n = 1.

Discussion

Introduction

In our previous article, we solved a system of equations using the substitution method. In this article, we will answer some frequently asked questions about solving systems of equations using the substitution method.

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of equations by substituting one equation into the other equation. This method is useful when one of the equations can be easily solved for one of the variables.

Q: How do I know which equation to solve for first?

A: To determine which equation to solve for first, look for the equation that can be easily solved for one of the variables. For example, if one of the equations has a variable with a coefficient of 1, it may be easier to solve for that variable first.

Q: What if I get stuck during the substitution process?

A: If you get stuck during the substitution process, try to simplify the equation by combining like terms or using algebraic properties. If you are still having trouble, try using a different method, such as the elimination method.

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

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

Q: What if the system of equations has no solution or an infinite number of solutions?

A: If the system of equations has no solution, it means that the equations are inconsistent and cannot be solved simultaneously. If the system of equations has an infinite number of solutions, it means that the equations are dependent and have the same solution.

Q: Can I use the substitution method with systems of equations that have fractions or decimals?

A: Yes, you can use the substitution method with systems of equations that have fractions or decimals. However, you may need to simplify the equations by multiplying both sides by a common denominator or using algebraic properties.

Q: What are some common mistakes to avoid when using the substitution method?

A: Some common mistakes to avoid when using the substitution method include:

  • Not simplifying the equations before substituting
  • Not checking for extraneous solutions
  • Not using algebraic properties to simplify the equations
  • Not checking the solution for consistency with the original equations

Conclusion

In this article, we answered some frequently asked questions about solving systems of equations using the substitution method. We discussed how to determine which equation to solve for first, how to simplify the equations, and how to avoid common mistakes. We also discussed how to use the substitution method with systems of equations that have more than two variables, fractions, or decimals.

Tips and Tricks

Here are some tips and tricks to help you master the substitution method:

  • Practice, practice, practice: The more you practice solving systems of equations using the substitution method, the more comfortable you will become with the process.
  • Use algebraic properties: Algebraic properties, such as the distributive property and the commutative property, can help you simplify the equations and make the substitution process easier.
  • Check for extraneous solutions: Make sure to check the solution for consistency with the original equations to avoid extraneous solutions.
  • Use a graphing calculator: If you are having trouble solving a system of equations using the substitution method, try using a graphing calculator to visualize the solution.

Conclusion

Solving systems of equations using the substitution method can be a powerful tool for solving mathematical problems. By following the steps outlined in this article and practicing regularly, you can master the substitution method and become proficient in solving systems of equations.