Make sure to solve one equation for a variable

Oct 13, 2025 by talynt.org 106 views

Introduction

Solving systems of equations is a fundamental concept in mathematics, and there are several methods to approach it. One of the most effective methods is substitution, which involves solving one equation for a variable and then substituting that expression into the other equation. In this article, we will explore how to solve a system of equations using the substitution method, with a focus on a specific example.

The Substitution Method

The substitution method is a powerful tool for solving systems of equations. It involves solving one equation for a variable and then substituting that expression into the other equation. This method is particularly useful when one of the equations is easily solvable for a variable.

Step 1: Solve One Equation for a Variable

To begin the substitution method, we need to solve one of the equations for a variable. In this case, we have the equation:

x = -3y

This equation is already solved for $x$ , so we can move on to the next step.

Step 2: Substitute the Expression into the Other Equation

Now that we have an expression for $x$ , we can substitute it into the other equation:

-6x - 8y = 50

Substituting $x = -3y$ into this equation, we get:

-6(-3y) - 8y = 50

Simplifying this equation, we get:

18y - 8y = 50

Combine like terms:

10y = 50

Step 3: Solve for the Variable

Now that we have a simplified equation, we can solve for the variable $y$ . Dividing both sides of the equation by 10, we get:

y = 5

Step 4: Find the Value of the Other Variable

Now that we have the value of $y$ , we can find the value of the other variable $x$ . Substituting $y = 5$ into the equation $x = -3y$ , we get:

x = -3(5)

Simplifying this equation, we get:

x = -15

Conclusion

In this article, we have explored how to solve a system of equations using the substitution method. We began by solving one equation for a variable, then substituted that expression into the other equation. We simplified the resulting equation and solved for the variable. Finally, we found the value of the other variable by substituting the value of the first variable into the equation. This method is a powerful tool for solving systems of equations, and it is particularly useful when one of the equations is easily solvable for a variable.

Example: Solving a System of Equations

Let's consider another example of a system of equations:

\begin{aligned} -6x - 8y &= 50 \\ x &= -3y \end{aligned}

To solve this system of equations, we can follow the same steps as before. First, we solve one equation for a variable:

x = -3y

Next, we substitute this expression into the other equation:

-6x - 8y = 50

Substituting $x = -3y$ into this equation, we get:

-6(-3y) - 8y = 50

Simplifying this equation, we get:

18y - 8y = 50

Combine like terms:

10y = 50

Solving for $y$ , we get:

y = 5

Finally, we find the value of the other variable $x$ by substituting $y = 5$ into the equation $x = -3y$ :

x = -3(5)

Simplifying this equation, we get:

x = -15

Tips and Tricks

Here are some tips and tricks to keep in mind when using the substitution method:

Make sure to solve one equation for a variable: This is the first step in the substitution method. Make sure to solve one equation for a variable before substituting it into the other equation.
Substitute the expression carefully: When substituting the expression into the other equation, make sure to substitute it correctly. Check your work to make sure that the equation is simplified correctly.
Solve for the variable: Once you have simplified the equation, solve for the variable. Make sure to check your work to make sure that the equation is solved correctly.
Find the value of the other variable: Finally, find the value of the other variable by substituting the value of the first variable into the equation.

Conclusion

In conclusion, the substitution method is a powerful tool for solving systems of equations. By solving one equation for a variable and then substituting that expression into the other equation, we can solve for the variables in a system of equations. This method is particularly useful when one of the equations is easily solvable for a variable. With practice and patience, you can master the substitution method and become proficient in solving systems of equations.

Frequently Asked Questions

Here are some frequently asked questions about the substitution method:

What is the substitution method?: The substitution method is a method for solving systems of equations by solving one equation for a variable and then substituting that expression into the other equation.
How do I use the substitution method?: To use the substitution method, solve one equation for a variable, substitute that expression into the other equation, simplify the resulting equation, and solve for the variable.
What are some tips and tricks for using the substitution method?: Some tips and tricks for using the substitution method include making sure to solve one equation for a variable, substituting the expression carefully, solving for the variable, and finding the value of the other variable.

References

Here are some references for further reading on the substitution method:

Algebra textbooks: Many algebra textbooks cover the substitution method in detail. Check your textbook for more information.
Online resources: There are many online resources available that cover the substitution method, including video tutorials and practice problems.
Math websites: Websites such as Khan Academy and Mathway offer video tutorials and practice problems on the substitution method.

Conclusion

Introduction

In our previous article, we explored how to solve systems of equations using the substitution method. This method is a powerful tool for solving systems of equations, and it is particularly useful when one of the equations is easily solvable for a variable. In this article, we will answer some frequently asked questions about the substitution method.

Q&A

Q: What is the substitution method?

A: The substitution method is a method for solving systems of equations by solving one equation for a variable and then substituting that expression into the other equation.

Q: How do I use the substitution method?

A: To use the substitution method, solve one equation for a variable, substitute that expression into the other equation, simplify the resulting equation, and solve for the variable.

Q: What are some tips and tricks for using the substitution method?

A: Some tips and tricks for using the substitution method include making sure to solve one equation for a variable, substituting the expression carefully, solving for the variable, and finding the value of the other variable.

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 solving one equation for a variable: Make sure to solve one equation for a variable before substituting it into the other equation.
Not substituting the expression carefully: Make sure to substitute the expression correctly into the other equation.
Not simplifying the resulting equation: Make sure to simplify the resulting equation before solving for the variable.
Not checking the solution: Make sure to check the solution to make sure that it is correct.

Q: Can I use the substitution method with any type of system of equations?

A: The substitution method can be used with any type of system of equations, including linear and nonlinear systems.

Q: How do I know if the substitution method is the best method to use?

A: The substitution method is a good choice when one of the equations is easily solvable for a variable. If one of the equations is not easily solvable for a variable, you may want to consider using a different method, such as the elimination method.

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

A: Yes, the substitution method can be used with systems of equations with more than two variables. However, it may be more difficult to solve the system of equations in this case.

Q: How do I check my solution to make sure that it is correct?

A: To check your solution, make sure that it satisfies both equations in the system of equations. You can do this by substituting the solution into both equations and checking that the equations are true.

Conclusion

Frequently Asked Questions

Here are some frequently asked questions about the substitution method:

What is the substitution method?
How do I use the substitution method?
What are some tips and tricks for using the substitution method?
What are some common mistakes to avoid when using the substitution method?
Can I use the substitution method with any type of system of equations?
How do I know if the substitution method is the best method to use?
Can I use the substitution method with systems of equations with more than two variables?
How do I check my solution to make sure that it is correct?

References

Here are some references for further reading on the substitution method:

Algebra textbooks: Many algebra textbooks cover the substitution method in detail. Check your textbook for more information.
Online resources: There are many online resources available that cover the substitution method, including video tutorials and practice problems.
Math websites: Websites such as Khan Academy and Mathway offer video tutorials and practice problems on the substitution method.