Solving Systems with Two Variables

Oct 13, 2025 by wbrum.org 119 views

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Introduction

In algebra, solving systems of linear equations is a crucial concept that helps us find the solution to a set of equations. One of the methods used to solve these systems is the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation. In this article, we will explore how to solve a system of linear equations using the substitution method, with a focus on the given system: $${ \begin{aligned} x & = -8y \ -x + y & = 18 \end{aligned} }$$

Understanding the System

Before we dive into solving the system, let's take a closer look at the given equations. We have two equations:

$x = -8y$
$-x + y = 18$

The first equation is already solved for $x$ , which is equal to $-8y$ . This means that we can substitute $-8y$ for $x$ in the second equation.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. In this case, we will substitute $-8y$ for $x$ in the second equation.

Step 1: Substitute $-8y$ for $x$ in the Second Equation

We will replace $x$ with $-8y$ in the second equation: $-x + y = 18$ . This gives us:

$-(-8y) + y = 18$

Simplifying the equation, we get:

$8y + y = 18$

Combine like terms:

$9y = 18$

Step 2: Solve for $y$

Now that we have the equation $9y = 18$ , we can solve for $y$ by dividing both sides by 9:

$y = \frac{18}{9}$

Simplifying the fraction, we get:

$y = 2$

Step 3: Substitute $y$ into One of the Original Equations

Now that we have the value of $y$ , we can substitute it into one of the original equations to find the value of $x$ . We will use the first equation: $x = -8y$ .

Substituting $y = 2$ into the equation, we get:

$x = -8(2)$

Simplifying the equation, we get:

$x = -16$

Conclusion

In this article, we solved a system of linear equations using the substitution method. We started by substituting $-8y$ for $x$ in the second equation, which gave us a new equation in terms of $y$ . We then solved for $y$ by dividing both sides of the equation by 9. Finally, we substituted the value of $y$ into one of the original equations to find the value of $x$ . The solution to the system is $x = -16$ and $y = 2$ .

Example Use Cases

The substitution method is a powerful tool for solving systems of linear equations. Here are a few example use cases:

Solving Systems with Two Variables: The substitution method is particularly useful when solving systems with two variables. By substituting one variable into the other equation, we can solve for the remaining variable.
Solving Systems with Multiple Equations: The substitution method can also be used to solve systems with multiple equations. By substituting one equation into another, we can eliminate variables and solve for the remaining variables.
Solving Systems with Non-Linear Equations: The substitution method can also be used to solve systems with non-linear equations. By substituting one equation into another, we can eliminate variables and solve for the remaining variables.

Tips and Tricks

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

Simplify the Equations: Before substituting one equation into another, make sure to simplify the equations as much as possible. This will make it easier to substitute and solve for the remaining variables.
Check Your Work: After solving the system, make sure to check your work by plugging the values back into the original equations. This will ensure that the solution is correct.
Use the Substitution Method with Caution: The substitution method can be a powerful tool, but it can also be tricky to use. Make sure to use it with caution and to check your work carefully.

Conclusion

In conclusion, the substitution method is a powerful tool for solving systems of linear equations. By substituting one equation into another, we can eliminate variables and solve for the remaining variables. With practice and patience, the substitution method can be a valuable tool in your algebra toolkit.

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Introduction

In our previous article, we explored how to solve a system of linear equations using the substitution method. In this article, we will answer some of the most frequently asked questions about solving systems by substitution.

Q&A

Q: What is the substitution method?

A: The substitution method is a technique used to solve systems of linear equations by substituting one equation into another. This method involves solving one equation for one variable and then substituting that expression 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 already solved for one variable. This method is particularly useful when solving systems with two 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 variable into the other equation. This will allow you to eliminate one variable and solve for the remaining variable.

Q: What if I have a system with multiple equations?

A: You can use the substitution method to solve systems with multiple equations. Simply substitute one equation into another, and then substitute the resulting equation into another equation, and so on.

Q: Can I use the substitution method to solve systems with non-linear equations?

A: Yes, you can use the substitution method to solve systems with non-linear equations. However, you may need to use additional techniques, such as factoring or the quadratic formula, to solve the resulting equation.

Q: How do I check my work when using the substitution method?

A: To check your work, plug the values back into the original equations. If the values satisfy both equations, then the solution is correct.

Q: What if I get stuck while using the substitution method?

A: If you get stuck, try simplifying the equations or using a different method, such as the elimination method. You can also try graphing the equations on a coordinate plane to see if they intersect.

Q: Can I use the substitution method to solve systems with fractions or decimals?

A: Yes, you can use the substitution method to solve systems with fractions or decimals. Simply substitute the fractions or decimals into the equations and solve as usual.

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 already solved for one variable. However, if the equations are complex or involve multiple variables, you may want to use a different method, such as the elimination method.

Example Problems

Here are a few example problems to help illustrate the substitution method:

Example 1

Solve the system:

{ \begin{aligned} x & = 3y \\ 2x - 5y & = -3 \end{aligned} \}

Substitute $x = 3y$ into the second equation:

$2(3y) - 5y = -3$

Simplify the equation:

$6y - 5y = -3$

Combine like terms:

$y = -3$

Substitute $y = -3$ into the first equation:

$x = 3(-3)$

Simplify the equation:

$x = -9$

The solution to the system is $x = -9$ and $y = -3$ .

Example 2

Solve the system:

{ \begin{aligned} y & = 2x \\ x + 3y & = 12 \end{aligned} \}

Substitute $y = 2x$ into the second equation:

$x + 3(2x) = 12$

Simplify the equation:

$x + 6x = 12$

Combine like terms:

$7x = 12$

Divide both sides by 7:

$x = \frac{12}{7}$

Substitute $x = \frac{12}{7}$ into the first equation:

$y = 2\left(\frac{12}{7}\right)$

Simplify the equation:

$y = \frac{24}{7}$

The solution to the system is $x = \frac{12}{7}$ and $y = \frac{24}{7}$ .