Solve This System Of Equations Using Substitution:$\[ \begin{array}{l} y = 2x - 4 \\ x + 2y = 10 \end{array} \\]Step 1: Substitute \[$ Y = 2x - 4 \$\] Into The Second Equation:$\[ x + 2(2x - 4) = 10 \\]Step 2: Simplify The

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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 a system of equations using the substitution method. This method involves substituting one equation into the other to solve for the unknown variables.

Step 1: Substitute the First Equation into the Second Equation

The given system of equations is:

{ \begin{array}{l} y = 2x - 4 \\ x + 2y = 10 \end{array} \}

To solve this system using substitution, we will substitute the first equation into the second equation. The first equation is y=2x−4y = 2x - 4. We will substitute this expression for yy into the second equation.

Step 1: Substitute y=2x−4y = 2x - 4 into the Second Equation

Substituting y=2x−4y = 2x - 4 into the second equation, we get:

{ x + 2(2x - 4) = 10 \}

This equation can be simplified by distributing the 2 to the terms inside the parentheses.

Step 2: Simplify the Equation

To simplify the equation, we will distribute the 2 to the terms inside the parentheses:

{ x + 4x - 8 = 10 \}

Combining like terms, we get:

{ 5x - 8 = 10 \}

Step 3: Add 8 to Both Sides of the Equation

To isolate the term with the variable, we will add 8 to both sides of the equation:

{ 5x - 8 + 8 = 10 + 8 \}

Simplifying the equation, we get:

{ 5x = 18 \}

Step 4: Divide Both Sides of the Equation by 5

To solve for xx, we will divide both sides of the equation by 5:

{ \frac{5x}{5} = \frac{18}{5} \}

Simplifying the equation, we get:

{ x = \frac{18}{5} \}

Step 5: Substitute the Value of xx into the First Equation

Now that we have found the value of xx, we can substitute it into the first equation to find the value of yy.

{ y = 2x - 4 \}

Substituting x=185x = \frac{18}{5} into the first equation, we get:

{ y = 2\left(\frac{18}{5}\right) - 4 \}

Simplifying the equation, we get:

{ y = \frac{36}{5} - 4 \}

Step 6: Simplify the Equation

To simplify the equation, we will convert the fraction to a decimal:

{ y = \frac{36}{5} - \frac{20}{5} \}

Simplifying the equation, we get:

{ y = \frac{16}{5} \}

Conclusion

In this article, we have solved a system of equations using the substitution method. We have substituted the first equation into the second equation, simplified the equation, and solved for the unknown variables. The final solution is x=185x = \frac{18}{5} and y=165y = \frac{16}{5}.

Discussion

The substitution method is a powerful tool for solving systems of equations. It involves substituting one equation into the other to solve for the unknown variables. This method is particularly useful when one of the equations is linear and the other is quadratic.

Example Problems

Here are some example problems that can be solved using the substitution method:

  1. Solve the system of equations:

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

  1. Solve the system of equations:

{ \begin{array}{l} y = 2x + 1 \\ x - 2y = -3 \end{array} \}

Tips and Tricks

Here are some tips and tricks for solving systems of equations using the substitution method:

  1. Make sure to substitute the correct equation into the other equation.
  2. Simplify the equation by combining like terms.
  3. Isolate the term with the variable by adding or subtracting the same value to both sides of the equation.
  4. Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Introduction

In our previous article, we discussed how to solve systems of equations using the substitution method. This method involves substituting one equation into the other to solve for the unknown variables. 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 systems of equations by substituting one equation into the other to solve for the unknown variables.

Q: When should I use the substitution method?

A: You should use the substitution method when one of the equations is linear and the other is quadratic, or when one of the equations is already solved for one of the variables.

Q: How do I know which equation to substitute into the other?

A: You should substitute the equation that is already solved for one of the variables into the other equation.

Q: What if I have two linear equations? Can I still use the substitution method?

A: Yes, you can still use the substitution method even if you have two linear equations. Simply substitute one equation into the other and solve for the unknown variables.

Q: What if I have a quadratic equation? Can I still use the substitution method?

A: Yes, you can still use the substitution method even if you have a quadratic equation. However, you may need to use the quadratic formula to solve for the unknown variables.

Q: How do I simplify the equation after substituting one equation into the other?

A: To simplify the equation, combine like terms and isolate the term with the variable by adding or subtracting the same value to both sides of the equation.

Q: What if I get a fraction or decimal as my answer? How do I simplify it?

A: If you get a fraction or decimal as your answer, you can simplify it by converting the fraction to a decimal or vice versa.

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

A: Yes, you can use the substitution method to solve systems of equations with more than two variables. However, you may need to use the method multiple times to solve for all the variables.

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:

  • Substituting the wrong equation into the other equation
  • Not simplifying the equation after substituting one equation into the other
  • Not isolating the term with the variable by adding or subtracting the same value to both sides of the equation
  • Not using the correct method to solve for the unknown variables

Conclusion

In this article, we have answered some frequently asked questions about solving systems of equations using the substitution method. We have discussed when to use the substitution method, how to simplify the equation after substituting one equation into the other, and how to avoid common mistakes. By following these tips and tricks, you can solve systems of equations using the substitution method with ease.

Additional Resources

Here are some additional resources that you can use to learn more about solving systems of equations using the substitution method:

  • Khan Academy: Solving Systems of Equations Using Substitution
  • Mathway: Solving Systems of Equations Using Substitution
  • Wolfram Alpha: Solving Systems of Equations Using Substitution

Practice Problems

Here are some practice problems that you can use to practice solving systems of equations using the substitution method:

  1. Solve the system of equations:

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

  1. Solve the system of equations:

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

  1. Solve the system of equations:

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

Answer Key

Here are the answers to the practice problems:

  1. x=2x = 2, y=1y = 1
  2. x=−1x = -1, y=0y = 0
  3. x=4x = 4, y=10y = 10