Solve Using Substitution.$\[ \begin{array}{l} -x - 6y = -16 \\ x = -8 \end{array} \\]

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear equations using the substitution method. This method involves substituting the value of one variable into the other equation to solve for the remaining variable. We will use a step-by-step approach to demonstrate how to solve linear equations using substitution.

What is Substitution Method?

The substitution method is a technique used to solve linear equations by substituting the value of one variable into the other equation. This method is useful when we have two linear equations with two variables, and we need to solve for one of the variables. The substitution method involves the following steps:

  1. Solve one of the equations for one variable: We need to solve one of the equations for one variable in terms of the other variable.
  2. Substitute the value of the variable into the other equation: We substitute the value of the variable into the other equation to solve for the remaining variable.
  3. Solve for the remaining variable: We solve for the remaining variable using the substituted equation.

Step-by-Step Solution

Let's use the given problem to demonstrate how to solve linear equations using substitution.

Problem

{ \begin{array}{l} -x - 6y = -16 \\ x = -8 \end{array} \}

Step 1: Solve one of the equations for one variable

We are given that x=βˆ’8x = -8. We can substitute this value into the first equation to solve for yy.

βˆ’xβˆ’6y=βˆ’16βˆ’(βˆ’8)βˆ’6y=βˆ’168βˆ’6y=βˆ’16{ -x - 6y = -16 \\ -(-8) - 6y = -16 \\ 8 - 6y = -16 }

Step 2: Substitute the value of the variable into the other equation

We can substitute x=βˆ’8x = -8 into the first equation to get:

βˆ’(βˆ’8)βˆ’6y=βˆ’168βˆ’6y=βˆ’16{ -(-8) - 6y = -16 \\ 8 - 6y = -16 }

Step 3: Solve for the remaining variable

We can solve for yy by isolating it on one side of the equation.

8βˆ’6y=βˆ’16βˆ’6y=βˆ’24y=4{ 8 - 6y = -16 \\ -6y = -24 \\ y = 4 }

Conclusion

In this article, we demonstrated how to solve linear equations using the substitution method. We used a step-by-step approach to solve the given problem, and we were able to find the value of the remaining variable. The substitution method is a useful technique for solving linear equations, and it can be applied to a wide range of problems.

Real-World Applications

The substitution method has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, the substitution method can be used to solve problems involving motion and forces. In engineering, the substitution method can be used to design and optimize systems. In economics, the substitution method can be used to analyze and predict market trends.

Tips and Tricks

Here are some tips and tricks to help you master the substitution method:

  • Make sure to read the problem carefully: Before starting to solve the problem, make sure to read it carefully and understand what is being asked.
  • Use the correct substitution: Make sure to substitute the correct value of the variable into the other equation.
  • Solve for the remaining variable: Make sure to solve for the remaining variable using the substituted equation.
  • Check your work: Make sure to check your work by plugging the values back into the original equations.

Common Mistakes

Here are some common mistakes to avoid when using the substitution method:

  • Not reading the problem carefully: Failing to read the problem carefully can lead to incorrect substitutions and solutions.
  • Using the wrong substitution: Using the wrong substitution can lead to incorrect solutions.
  • Not solving for the remaining variable: Failing to solve for the remaining variable can lead to incomplete solutions.
  • Not checking your work: Failing to check your work can lead to incorrect solutions.

Conclusion

Q: What is the substitution method?

A: The substitution method is a technique used to solve linear equations by substituting the value of one variable 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 with two variables, and you need to solve for one of the variables.

Q: How do I know which variable to substitute first?

A: You should substitute the variable that is easiest to solve for first. In the given problem, we were given that x=βˆ’8x = -8, so we substituted that value into the first equation.

Q: What if I have two equations with two variables, but I don't have a value for either variable?

A: In that case, you can use the substitution method to solve for one of the variables, and then use the other equation to solve for the remaining variable.

Q: Can I use the substitution method to solve quadratic equations?

A: No, the substitution method is only used to solve linear equations. If you have a quadratic equation, you will need to use a different method, such as factoring or the quadratic formula.

Q: What if I make a mistake while using the substitution method?

A: If you make a mistake while using the substitution method, you may end up with an incorrect solution. To avoid this, make sure to read the problem carefully, use the correct substitution, solve for the remaining variable, and check your work.

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

A: No, the substitution method is only used to solve systems of equations with two variables. If you have a system of equations with more than two variables, you will need to use a different method, such as the elimination method or the matrix method.

Q: How do I know if I have solved the equation correctly?

A: To check if you have solved the equation correctly, plug the values back into the original equations and make sure they are true.

Q: What if I get a negative value for one of the variables?

A: If you get a negative value for one of the variables, it is still a valid solution. Just make sure to include the negative sign when writing the final answer.

Q: Can I use the substitution method to solve equations with fractions?

A: Yes, you can use the substitution method to solve equations with fractions. Just make sure to simplify the fractions before substituting the value of the variable.

Q: What if I have a system of equations with variables that have exponents?

A: If you have a system of equations with variables that have exponents, you will need to use a different method, such as the elimination method or the matrix method.

Conclusion

In conclusion, the substitution method is a powerful technique for solving linear equations. By following the steps outlined in this article, you can master the substitution method and apply it to a wide range of problems. Remember to read the problem carefully, use the correct substitution, solve for the remaining variable, and check your work. With practice and patience, you can become proficient in using the substitution method to solve linear equations.