Use Substitution To Solve The System Of Equations:${ \begin{align} 7x + 3y &= 1 \ x - 2y &= -12 \end{align} }$

Mar 13, 2025 by ADMIN 114 views

**Solving Systems of Equations Using Substitution Method**

Introduction to Substitution Method

The substitution method is a technique used to solve systems of linear equations by expressing one variable in terms of the other and then substituting it into the other equation. This method is particularly useful when one of the equations is easily solvable for one of the variables. In this article, we will use the substitution method to solve the given system of equations:

{ \begin{align*} 7x + 3y &= 1 \\ x - 2y &= -12 \end{align*} \}

Step 1: Solve One Equation for One Variable

To start the substitution method, we need to solve one of the equations for one of the variables. Let's solve the second equation for x:

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

Step 2: Substitute the Expression into the Other Equation

Now that we have expressed x in terms of y, we can substitute this expression into the first equation:

{ \begin{align*} 7x + 3y &= 1 \\ 7(-12 + 2y) + 3y &= 1 \end{align*} \}

Step 3: Simplify the Equation

Next, we simplify the equation by distributing the 7 and combining like terms:

{ \begin{align*} -84 + 14y + 3y &= 1 \\ -84 + 17y &= 1 \end{align*} \}

Step 4: Solve for the Variable

Now, we can solve for y by isolating it on one side of the equation:

{ \begin{align*} -84 + 17y &= 1 \\ 17y &= 1 + 84 \\ 17y &= 85 \\ y &= \frac{85}{17} \end{align*} \}

Step 5: Find the Value of the Other Variable

Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use the second equation:

{ \begin{align*} x - 2y &= -12 \\ x - 2\left(\frac{85}{17}\right) &= -12 \\ x - \frac{170}{17} &= -12 \\ x &= -12 + \frac{170}{17} \\ x &= \frac{-204 + 170}{17} \\ x &= \frac{-34}{17} \end{align*} \}

Conclusion

In this article, we used the substitution method to solve the given system of equations. We first solved one of the equations for one of the variables, then substituted the expression into the other equation, simplified the equation, solved for the variable, and finally found the value of the other variable. The substitution method is a powerful tool for solving systems of linear equations, and it is particularly useful when one of the equations is easily solvable for one of the variables.

Example Problems

Here are a few example problems that you can try using the substitution method:

\begin{align*} 2x + 5y &= 11 \ x - 3y &= -7 \end{align*} }$

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

\begin{align*} x + 2y &= 6 \ 2x - 3y &= 9 \end{align*} }$

Tips and Tricks

Here are a few tips and tricks to help you use the substitution method effectively:

Make sure to solve one of the equations for one of the variables before substituting it into the other equation.
Simplify the equation as much as possible before solving for the variable.
Check your work by plugging the values back into the original equations.
Practice, practice, practice! The more you practice using the substitution method, the more comfortable you will become with it.

Real-World Applications

The substitution method has many real-world applications, including:

Physics and Engineering: The substitution method is used to solve systems of equations that describe the motion of objects in physics and engineering.
Computer Science: The substitution method is used to solve systems of equations that describe the behavior of computer systems and networks.
Economics: The substitution method is used to solve systems of equations that describe the behavior of economic systems and markets.

Conclusion

In conclusion, the substitution method is a powerful tool for solving systems of linear equations. It is particularly useful when one of the equations is easily solvable for one of the variables. By following the steps outlined in this article, you can use the substitution method to solve a wide range of systems of equations. With practice and patience, you will become proficient in using the substitution method and be able to apply it to a variety of real-world problems.

Q: What is the substitution method?

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

Q: When should I use the substitution method?

A: You should use the substitution method when one of the equations is easily solvable for one of the variables. This method is particularly useful when you can easily express one variable in terms of the other.

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

A: You can choose either equation to solve first, but it's often easier to solve the equation that has the variable you want to express in terms of the other variable.

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

A: If you get stuck, try simplifying the equation or checking your work to make sure you haven't made any mistakes. You can also try using a different method, such as the elimination 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: How do I check my work when using the substitution method?

A: To check your work, plug the values you found back into the original equations to make sure they are true. If the values satisfy both equations, then you have found the correct solution.

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 equation enough before solving for the variable
Not checking your work to make sure the values satisfy both equations
Not using the correct method for the type of system of equations you are working with

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 equation more carefully to avoid mistakes.

Q: How do I know if the substitution method is the best method to use for a particular system of equations?

A: To determine if the substitution method is the best method to use, try solving one of the equations for one of the variables. If you can easily express one variable in terms of the other, then the substitution method may be the best choice.

Q: Can I use the substitution method with systems of equations that have negative numbers?

A: Yes, you can use the substitution method with systems of equations that have negative numbers. However, you may need to be careful when simplifying the equation to avoid mistakes.

Q: How do I extend the substitution method to solve systems of equations with more than two variables?

A: To extend the substitution method to solve systems of equations with more than two variables, you can use the same steps as before, but you will need to express one variable in terms of the other variables and then substitute it into the other equations.

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

A: Yes, you can use the substitution method with systems of equations that have variables with exponents. However, you may need to simplify the equation more carefully to avoid mistakes.

Q: How do I know if the substitution method is the best method to use for a particular system of equations with variables with exponents?

Conclusion

In conclusion, the substitution method is a powerful tool for solving systems of linear equations. By following the steps outlined in this article and avoiding common mistakes, you can use the substitution method to solve a wide range of systems of equations. With practice and patience, you will become proficient in using the substitution method and be able to apply it to a variety of real-world problems.