Solve Using Substitution.$\[ \begin{align*} y &= -10 \\ -10x - 9y &= -10 \end{align*} \\]

Introduction

In algebra, solving linear equations is a fundamental concept that forms the basis of more complex mathematical operations. One of the methods used to solve linear equations is the substitution method. This method involves substituting the value of one variable from one equation into the other equation to solve for the remaining variable. In this article, we will explore how to solve linear equations using the substitution method, with a focus on a specific example.

The Substitution Method

The substitution method is a step-by-step process that involves the following steps:

1. Identify the equations: Identify the two linear equations that need to be solved.
2. Solve one equation for one variable: Solve one of the equations for one of the variables.
3. Substitute the value into the other equation: Substitute the value of the variable from the first equation into the second equation.
4. Solve for the remaining variable: Solve the resulting equation for the remaining variable.

Example: Solving the Given Equations

Let's consider the following example:

{ \begin{align*} y &= -10 \\ -10x - 9y &= -10 \end{align*} \}

In this example, we have two linear equations:

1. $y = -10$
2. $-10x - 9y = -10$

We will use the substitution method to solve for the variable $x$.

Step 1: Solve One Equation for One Variable

We will start by solving the first equation for the variable $y$.

$y = -10$

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

Step 2: Substitute the Value into the Other Equation

Now, we will substitute the value of $y$ from the first equation into the second equation.

$-10x - 9y = -10$

Substituting $y = -10$ into the second equation, we get:

$-10x - 9(-10) = -10$

Step 3: Solve for the Remaining Variable

Now, we will solve the resulting equation for the remaining variable $x$.

$-10x + 90 = -10$

Subtracting 90 from both sides of the equation, we get:

$-10x = -100$

Dividing both sides of the equation by -10, we get:

$x = 10$

Therefore, the value of the variable $x$ is 10.

Conclusion

In this article, we explored how to solve linear equations using the substitution method. We used a specific example to illustrate the step-by-step process involved in solving linear equations using the substitution method. By following the steps outlined in this article, you can solve linear equations using the substitution method and develop a deeper understanding of algebraic operations.

Common Mistakes to Avoid

When solving linear equations using the substitution method, there are several common mistakes to avoid:

• Not solving one equation for one variable: Make sure to solve one equation for one variable before substituting the value into the other equation.
• Not substituting the value correctly: Make sure to substitute the value of the variable from the first equation into the second equation correctly.
• Not solving for the remaining variable: Make sure to solve the resulting equation for the remaining variable.

Real-World Applications

The substitution method is used in a variety of real-world applications, including:

• Physics: The substitution method is used to solve equations that describe the motion of objects.
• Engineering: The substitution method is used to solve equations that describe the behavior of electrical circuits.
• Computer Science: The substitution method is used to solve equations that describe the behavior of algorithms.

Final Thoughts

In conclusion, the substitution method is a powerful tool for solving linear equations. By following the steps outlined in this article, you can develop a deeper understanding of algebraic operations and solve linear equations using the substitution method. Remember to avoid common mistakes and apply the substitution method to real-world problems.

Additional Resources

For additional resources on solving linear equations using the substitution method, check out the following:

• Algebra textbooks: Check out algebra textbooks for additional examples and exercises on solving linear equations using the substitution method.
• Online resources: Check out online resources, such as Khan Academy and Mathway, for additional examples and exercises on solving linear equations using the substitution method.
• Practice problems: Practice solving linear equations using the substitution method with practice problems from online resources or algebra textbooks.
  Frequently Asked Questions (FAQs) on Solving Linear Equations Using the Substitution Method =====================================================================================

Q: What is the substitution method?

A: The substitution method is a step-by-step process used to solve linear equations by substituting the value of one variable from one equation into the other equation.

Q: When should I use the substitution method?

A: You should use the substitution method when you have two linear equations and you want to solve for one of the variables.

Q: How do I know which variable to solve for first?

A: You can solve for either variable first, but it's often easier to solve for the variable that appears in only one of the equations.

Q: What if I have multiple variables in the equation?

A: If you have multiple variables in the equation, you can use the substitution method to solve for one variable at a time.

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

A: No, the substitution method is only used with linear equations. If you have a non-linear equation, you may need to use a different method, such as the quadratic formula.

Q: What if I get a negative value for the variable?

A: If you get a negative value for the variable, it's still a valid solution. Just make sure to check your work to ensure that the solution is correct.

Q: Can I use the substitution method with equations that have fractions?

A: Yes, you can use the substitution method with equations that have fractions. Just make sure to simplify the fractions before substituting the value into the other equation.

Q: What if I get a decimal value for the variable?

A: If you get a decimal value for the variable, it's still a valid solution. Just make sure to check your work to ensure that the solution is correct.

Q: Can I use the substitution method with equations that have variables on both sides?

A: Yes, you can use the substitution method with equations that have variables on both sides. Just make sure to isolate the variable on one side of the equation before substituting the value into the other equation.

Q: What if I get a complex solution for the variable?

A: If you get a complex solution for the variable, it's still a valid solution. Just make sure to check your work to ensure that the solution is correct.

Q: Can I use the substitution method with equations that have absolute values?

A: Yes, you can use the substitution method with equations that have absolute values. Just make sure to simplify the absolute value expression before substituting the value into the other equation.

Q: What if I get a solution that doesn't satisfy the original equation?

A: If you get a solution that doesn't satisfy the original equation, it's not a valid solution. Make sure to check your work to ensure that the solution is correct.

Conclusion

In this article, we answered some of the most frequently asked questions about solving linear equations using the substitution method. We covered topics such as when to use the substitution method, how to choose which variable to solve for first, and how to handle equations with fractions, decimals, and absolute values. By following the steps outlined in this article, you can develop a deeper understanding of algebraic operations and solve linear equations using the substitution method.

Additional Resources

For additional resources on solving linear equations using the substitution method, check out the following:

• Algebra textbooks: Check out algebra textbooks for additional examples and exercises on solving linear equations using the substitution method.
• Online resources: Check out online resources, such as Khan Academy and Mathway, for additional examples and exercises on solving linear equations using the substitution method.
• Practice problems: Practice solving linear equations using the substitution method with practice problems from online resources or algebra textbooks.