Solve The System Using Substitution.${ \begin{array}{l} x = -2y - 6 \ 6x + 2y = 4 \ \end{array} }$([?], \square)

Mar 9, 2025 by ADMIN 114 views

**Solving Systems of Linear Equations using Substitution**

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. In this article, we will focus on solving systems of linear equations using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation to solve for the remaining variable.

What is the Substitution Method?

The substitution method is a technique used to solve systems of linear equations by substituting the expression for one variable from one equation into the other equation. This method is particularly useful when one of the equations is already solved for one variable. The substitution method involves the following steps:

Solve one equation for one variable.
Substitute the expression for the variable into the other equation.
Solve the resulting equation for the remaining variable.

Step-by-Step Guide to Solving Systems using Substitution

Step 1: Solve One Equation for One Variable

To solve the system using substitution, we need to solve one equation for one variable. Let's consider the given system of linear equations:

{ \begin{array}{l} x = -2y - 6 \\ 6x + 2y = 4 \\ \end{array} \}

We can solve the first equation for x:

$x = -2y - 6$

Step 2: Substitute the Expression into the Other Equation

Now that we have solved the first equation for x, we can substitute the expression into the second equation:

$6x + 2y = 4$

Substituting $x = -2y - 6$ into the second equation, we get:

$6(-2y - 6) + 2y = 4$

Step 3: Solve the Resulting Equation for the Remaining Variable

Now we need to solve the resulting equation for y:

$6(-2y - 6) + 2y = 4$

Expanding the equation, we get:

$-12y - 36 + 2y = 4$

Combine like terms:

$-10y - 36 = 4$

Add 36 to both sides:

$-10y = 40$

Divide both sides by -10:

$y = -4$

Step 4: 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 the other variable. Let's use the first equation:

$x = -2y - 6$

Substituting y = -4, we get:

$x = -2(-4) - 6$

Simplifying the equation, we get:

$x = 8 - 6$

$x = 2$

Conclusion

In this article, we have learned how to solve systems of linear equations using the substitution method. We have seen how to solve one equation for one variable and then substitute that expression into the other equation to solve for the remaining variable. We have also seen how to use the substitution method to solve a system of linear equations with two variables. The substitution method is a powerful tool for solving systems of linear equations, and it has numerous applications in various fields.

Example Problems

Example 1

Solve the system of linear equations using substitution:

{ \begin{array}{l} x + 2y = 6 \\ 3x - 2y = -3 \\ \end{array} \}

Solution

To solve the system using substitution, we need to solve one equation for one variable. Let's solve the first equation for x:

$x = 6 - 2y$

Substituting $x = 6 - 2y$ into the second equation, we get:

$3(6 - 2y) - 2y = -3$

Expanding the equation, we get:

$18 - 6y - 2y = -3$

Combine like terms:

$-8y + 18 = -3$

Subtract 18 from both sides:

$-8y = -21$

Divide both sides by -8:

$y = \frac{21}{8}$

Substituting y = 21/8 into the first equation, we get:

$x + 2(\frac{21}{8}) = 6$

Simplifying the equation, we get:

$x + \frac{21}{4} = 6$

Subtract 21/4 from both sides:

$x = 6 - \frac{21}{4}$

Simplifying the equation, we get:

$x = \frac{24 - 21}{4}$

$x = \frac{3}{4}$

Example 2

Solve the system of linear equations using substitution:

{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \\ \end{array} \}

Solution

To solve the system using substitution, we need to solve one equation for one variable. Let's solve the second equation for x:

$x = -3 + 2y$

Substituting $x = -3 + 2y$ into the first equation, we get:

$2(-3 + 2y) + 3y = 7$

Expanding the equation, we get:

$-6 + 4y + 3y = 7$

Combine like terms:

$7y - 6 = 7$

Add 6 to both sides:

$7y = 13$

Divide both sides by 7:

$y = \frac{13}{7}$

Substituting y = 13/7 into the second equation, we get:

$x - 2(\frac{13}{7}) = -3$

Simplifying the equation, we get:

$x - \frac{26}{7} = -3$

Add 26/7 to both sides:

$x = -3 + \frac{26}{7}$

Simplifying the equation, we get:

$x = \frac{-21 + 26}{7}$

$x = \frac{5}{7}$

Common Mistakes to Avoid

When solving systems of linear equations using substitution, 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 it into the other equation.
Not substituting the expression correctly: Make sure to substitute the expression for the variable into the other equation correctly.
Not simplifying the equation: Make sure to simplify the equation after substituting the expression.
Not checking the solution: Make sure to check the solution by substituting it back into both original equations.

Conclusion

Q: What is the substitution method?

A: The substitution method is a technique used to solve systems of linear equations by substituting the expression for 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 one of the equations is already solved for one variable. This method is particularly useful when you have a system of linear equations with two variables.

Q: How do I solve a system of linear equations using substitution?

A: To solve a system of linear equations using substitution, follow these steps:

Solve one equation for one variable.
Substitute the expression for the variable into the other equation.
Solve the resulting equation for the remaining 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 one variable.
Not substituting the expression correctly.
Not simplifying the equation.
Not checking the solution.

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

A: No, the substitution method is only suitable for solving systems of linear equations with two variables. If you have a system of linear equations with more than two variables, you may need to use other methods such as the elimination method or the matrix method.

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

A: To check your solution when using the substitution method, substitute the values of the variables back into both original equations and make sure they are true.

Q: What are some real-world applications of the substitution method?

A: The substitution method has numerous real-world applications in various fields such as physics, engineering, economics, and computer science. Some examples include:

Solving systems of linear equations to model population growth.
Solving systems of linear equations to determine the trajectory of a projectile.
Solving systems of linear equations to optimize business decisions.

Q: Can I use the substitution method to solve a system of linear equations with fractions or decimals?

A: Yes, you can use the substitution method to solve a system of linear equations with fractions or decimals. However, you may need to simplify the equations and perform calculations carefully to avoid errors.

Q: How do I choose between the substitution method and the elimination method?

A: The choice between the substitution method and the elimination method depends on the specific system of linear equations and your personal preference. If one equation is already solved for one variable, the substitution method may be more suitable. If the coefficients of the variables are easy to eliminate, the elimination method may be more suitable.

Q: Can I use the substitution method to solve a system of linear equations with complex numbers?

A: Yes, you can use the substitution method to solve a system of linear equations with complex numbers. However, you may need to perform calculations carefully to avoid errors and simplify the equations.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of linear equations using the substitution method. We have seen how to use the substitution method to solve systems of linear equations with two variables and how to check the solution. We have also seen some common mistakes to avoid and some real-world applications of the substitution method.