Solve The System Of Equations Using Substitution.${ \begin{align} x &= 3y - 6 \ 2x - 4y &= 8 \end{align} }$A. (3, -2) B. (10, 24) C. (24, 10) D. (-2, 0)

Mar 12, 2025 by ADMIN 159 views

Solving the System of Equations Using Substitution

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Introduction

Solving a system of equations using substitution is a powerful technique used in mathematics to find the solution to a set of linear equations. This method involves expressing one variable in terms of another variable from one equation and then substituting it into the other equation to solve for the remaining variable. In this article, we will explore how to solve a system of equations using substitution and apply it to a specific problem.

What is a System of Equations?

A system of equations is a set of two or more equations that contain two or more variables. Each equation is a statement that two expressions are equal, and the goal is to find the values of the variables that make all the equations true. Systems of equations can be solved using various methods, including substitution, elimination, and graphing.

Substitution Method

The substitution method involves expressing one variable in terms of another variable from one equation and then substituting it into the other equation to solve for the remaining variable. This method is particularly useful when one of the equations is already solved for one of the variables.

Step 1: Identify the Equation to Substitute

In the given problem, we have two equations:

{ \begin{align*} x &= 3y - 6 \\ 2x - 4y &= 8 \end{align*} \}

We can see that the first equation is already solved for $x$ , which makes it a good candidate for substitution.

Step 2: Substitute the Expression into the Other Equation

We will substitute the expression for $x$ from the first equation into the second equation:

{ \begin{align*} 2(3y - 6) - 4y &= 8 \end{align*} \}

Simplifying the equation, we get:

{ \begin{align*} 6y - 12 - 4y &= 8 \end{align*} \}

Combining like terms, we get:

{ \begin{align*} 2y - 12 &= 8 \end{align*} \}

Step 3: Solve for the Variable

Now, we can solve for $y$ by adding 12 to both sides of the equation:

{ \begin{align*} 2y &= 20 \end{align*} \}

Dividing both sides by 2, we get:

{ \begin{align*} y &= 10 \end{align*} \}

Step 4: Find the Value of the Other Variable

Now that we have the value of $y$ , we can substitute it back into one of the original equations to find the value of the other variable. We will use the first equation:

{ \begin{align*} x &= 3y - 6 \\ x &= 3(10) - 6 \\ x &= 30 - 6 \\ x &= 24 \end{align*} \}

Therefore, the solution to the system of equations is $x = 24$ and $y = 10$ .

Conclusion

Solving a system of equations using substitution is a powerful technique that can be used to find the solution to a set of linear equations. By expressing one variable in terms of another variable from one equation and then substituting it into the other equation, we can solve for the remaining variable. In this article, we applied the substitution method to a specific problem and found the solution to be $x = 24$ and $y = 10$ .

Discussion

The substitution method is a useful technique for solving systems of equations, but it requires careful attention to detail to ensure that the correct values are substituted into the equations. Additionally, the substitution method may not always be the most efficient method for solving systems of equations, especially when the equations are complex or have multiple variables.

Example Problems

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

\begin{align*} x + y &= 5 \ 2x - 3y &= 7 \end{align*} }$

\begin{align*} x - 2y &= 3 \ 3x + 4y &= 5 \end{align*} }$

\begin{align*} 2x + 3y &= 7 \ x - 4y &= -3 \end{align*} }$

These problems can be solved using the substitution method, and the solutions can be found by following the steps outlined in this article.

Final Answer

The final answer to the problem is:

$\boxed{(24, 10)}$

This is the solution to the system of equations using the substitution method.

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Introduction

In our previous article, we explored how to solve a system of equations using substitution. This method involves expressing one variable in terms of another variable from one equation and then substituting it into the other equation to solve for the remaining variable. In this article, we will answer some frequently asked questions about solving systems of equations using substitution.

Q&A

Q: What is the substitution method?

A: The substitution method is a technique used to solve systems of equations by expressing one variable in terms of another variable from one equation and then substituting it into the other equation to solve for the remaining variable.

Q: When should I use the substitution method?

A: You should use the substitution method when one of the equations is already solved for one of the variables. This makes it easier to substitute the expression into the other equation.

Q: How do I know which equation to substitute?

A: You should choose the equation that is already solved for one of the variables. This will make it easier to substitute the expression into the other equation.

Q: What if I have two equations with two variables, but neither equation is solved for one of the variables?

A: In this case, you can use the elimination method or the graphing method to solve the system of equations.

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, it may be more complicated and require more steps.

Q: What if I make a mistake while solving the system of equations using substitution?

A: If you make a mistake while solving the system of equations using substitution, you may end up with an incorrect solution. To avoid this, make sure to double-check your work and follow the steps carefully.

Q: Can I use a calculator to solve systems of equations using substitution?

A: Yes, you can use a calculator to solve systems of equations using substitution. However, it's always a good idea to check your work by hand to make sure you understand the solution.

Common Mistakes

When solving systems of equations using substitution, there are several common mistakes to watch out for:

Not following the steps carefully: Make sure to follow the steps carefully and double-check your work.
Not substituting the expression correctly: Make sure to substitute the expression correctly into the other equation.
Not solving for the correct variable: Make sure to solve for the correct variable.
Not checking the solution: Make sure to check the solution to make sure it satisfies both equations.

Tips and Tricks

Here are some tips and tricks to help you solve systems of equations using substitution:

Use a systematic approach: Use a systematic approach to solve the system of equations, such as following the steps outlined in this article.
Check your work: Make sure to check your work to make sure you understand the solution.
Use a calculator: You can use a calculator to solve systems of equations using substitution, but make sure to check your work by hand.
Practice, practice, practice: The more you practice solving systems of equations using substitution, the more comfortable you will become with the method.

Conclusion

Solving systems of equations using substitution is a powerful technique that can be used to find the solution to a set of linear equations. By expressing one variable in terms of another variable from one equation and then substituting it into the other equation, we can solve for the remaining variable. In this article, we answered some frequently asked questions about solving systems of equations using substitution and provided some tips and tricks to help you solve systems of equations using substitution.

Final Answer

The final answer to the problem is:

$\boxed{(24, 10)}$

This is the solution to the system of equations using the substitution method.