Solve This System Of Equations Using Substitution:$ \begin{array}{l} y = 2x - 4 \ x + 2y = 10 \end{array} }$Step 1 Substitute For { Y $ $ In The Second Equation. Which Is The Resulting Equation?A. { 2x - 4 = 10 $}$B.

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 substitution, a method that involves substituting one equation into another to eliminate one of the variables. We will use a specific example to illustrate this method and provide step-by-step instructions.

Step 1: Identify the Equations

The given system of equations is:

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

We have two equations: the first one is a linear equation in terms of $y$, and the second one is a linear equation in terms of $x$ and $y$.

Step 2: Substitute for $y$ in the Second Equation

To solve this system of equations using substitution, we need to substitute the expression for $y$ from the first equation into the second equation. This will eliminate the variable $y$ and allow us to solve for $x$.

The expression for $y$ is:

$$y = 2x - 4$$

We will substitute this expression into the second equation:

$$x + 2y = 10$$

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

$$x + 2(2x - 4) = 10$$

Step 3: Simplify the Resulting Equation

Now, we need to simplify the resulting equation:

$$x + 2(2x - 4) = 10$$

Using the distributive property, we can expand the equation:

$$x + 4x - 8 = 10$$

Combining like terms, we get:

$$5x - 8 = 10$$

Step 4: Solve for $x$

Now, we need to solve for $x$. We can add 8 to both sides of the equation to isolate the term with $x$:

$$5x - 8 + 8 = 10 + 8$$

This simplifies to:

$$5x = 18$$

Dividing both sides of the equation by 5, we get:

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

Step 5: Find the Value of $y$

Now that we have the value of $x$, we can find the value of $y$ by substituting $x$ into one of the original equations. We will use the first equation:

$$y = 2x - 4$$

Substituting $x = \frac{18}{5}$, we get:

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

Simplifying the equation, we get:

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

Using a common denominator, we can rewrite the equation as:

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

This simplifies to:

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

Conclusion

In this article, we solved a system of equations using substitution. We identified the equations, substituted for $y$ in the second equation, simplified the resulting equation, solved for $x$, and found the value of $y$. This method is a powerful tool for solving systems of equations, and it is essential to practice and master it.

Discussion

• What are some common mistakes to avoid when solving systems of equations using substitution?
• How can you determine which variable to substitute first?
• What are some real-world applications of solving systems of equations using substitution?

Answer Key

A. $x + 2(2x - 4) = 10$

Additional Resources

• Khan Academy: Solving Systems of Equations Using Substitution
• Mathway: Solving Systems of Equations Using Substitution
• Wolfram Alpha: Solving Systems of Equations Using Substitution
  Solving Systems of Equations Using Substitution: Q&A =====================================================

Introduction

In our previous article, we solved a system of equations using substitution. In this article, we will answer some frequently asked questions about solving systems of equations using substitution. Whether you are a student or a professional, this article will provide you with a deeper understanding of this method and help you to overcome common challenges.

Q: What are some common mistakes to avoid when solving systems of equations using substitution?

A: When solving systems of equations using substitution, there are several common mistakes to avoid:

• Not checking the validity of the solution: Before substituting the expression for $y$ into the second equation, make sure that the solution is valid. For example, if the solution involves dividing by zero, it is not valid.
• Not simplifying the resulting equation: After substituting the expression for $y$ into the second equation, make sure to simplify the resulting equation. This will help you to avoid errors and make it easier to solve for $x$.
• Not checking for extraneous solutions: When solving for $x$, make sure to check for extraneous solutions. An extraneous solution is a solution that is not valid for the original system of equations.

Q: How can you determine which variable to substitute first?

A: When solving systems of equations using substitution, you need to determine which variable to substitute first. Here are some tips to help you:

• Look for the variable with the simplest expression: If one variable has a simpler expression than the other, it is usually easier to substitute that variable first.
• Look for the variable with the most information: If one variable has more information than the other, it is usually easier to substitute that variable first.
• Try substituting both variables: If you are not sure which variable to substitute first, try substituting both variables and see which one works better.

Q: What are some real-world applications of solving systems of equations using substitution?

A: Solving systems of equations using substitution has many real-world applications, including:

• Physics and engineering: Solving systems of equations is essential in physics and engineering, where you need to solve for multiple variables to describe a physical system.
• Economics: Solving systems of equations is essential in economics, where you need to solve for multiple variables to describe economic systems.
• Computer science: Solving systems of equations is essential in computer science, where you need to solve for multiple variables to describe complex systems.

Q: How can you check for extraneous solutions?

A: When solving for $x$, you need to check for extraneous solutions. Here are some tips to help you:

• Plug the solution back into the original equations: After solving for $x$, plug the solution back into the original equations to check if it is valid.
• Check for division by zero: Make sure that the solution does not involve dividing by zero.
• Check for inconsistent equations: Make sure that the solution does not involve inconsistent equations.

Q: What are some common types of systems of equations that can be solved using substitution?

A: There are several common types of systems of equations that can be solved using substitution, including:

• Linear systems of equations: Linear systems of equations involve linear equations with no quadratic terms.
• Quadratic systems of equations: Quadratic systems of equations involve quadratic equations with no linear terms.
• Systems of equations with multiple variables: Systems of equations with multiple variables involve multiple variables and multiple equations.

Conclusion

In this article, we answered some frequently asked questions about solving systems of equations using substitution. Whether you are a student or a professional, this article will provide you with a deeper understanding of this method and help you to overcome common challenges.

Discussion

• What are some other common mistakes to avoid when solving systems of equations using substitution?
• How can you determine which variable to substitute first in a system of equations with multiple variables?
• What are some other real-world applications of solving systems of equations using substitution?

Answer Key

• Not checking the validity of the solution
• Not simplifying the resulting equation
• Not checking for extraneous solutions

Additional Resources

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