Solve The System By Substitution.${ \begin{array}{l} 2x - 5y = -9 \ y = 3 \end{array} }$Select The Correct Response:A. (3, 6)B. (3, 3)C. (1, 3)D. (2, 1)

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations is a fundamental concept in algebra and is used to find the values of the variables that satisfy all the equations in the system. There are several methods to solve a system of linear equations, including substitution, elimination, and graphing. In this article, we will focus on solving a system of linear equations by substitution.

What is Substitution Method?

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

1. Solve one of the equations for one of the variables.
2. Substitute the expression for the variable into the other equation.
3. Solve the resulting equation for the other variable.
4. Substitute the value of the second variable back into one of the original equations to find the value of the first variable.

Step-by-Step Solution

Let's consider the following system of linear equations:

{ \begin{array}{l} 2x - 5y = -9 \\ y = 3 \end{array} \}

We are given that $y = 3$. To solve for $x$, we can substitute $y = 3$ into the first equation.

Substituting y = 3 into the First Equation

Substituting $y = 3$ into the first equation, we get:

$2x - 5(3) = -9$

Simplifying the equation, we get:

$2x - 15 = -9$

Adding 15 to both sides of the equation, we get:

$2x = 6$

Dividing both sides of the equation by 2, we get:

$x = 3$

Finding the Value of y

We are already given that $y = 3$. Therefore, the solution to the system of linear equations is $(x, y) = (3, 3)$.

Conclusion

In this article, we solved a system of linear equations by substitution. We used the substitution method to find the values of the variables that satisfy both equations in the system. The substitution method is a useful technique for solving systems of linear equations, especially when one of the equations is already solved for one of the variables.

Answer

The correct answer is:

• B. (3, 3)

Discussion

Solving a system of linear equations by substitution is a fundamental concept in algebra. The substitution method is a useful technique for solving systems of linear equations, especially when one of the equations is already solved for one of the variables. In this article, we used the substitution method to solve a system of linear equations and found the values of the variables that satisfy both equations in the system.

Related Topics

• Solving a system of linear equations by elimination
• Solving a system of linear equations by graphing
• Systems of linear equations with three variables
• Nonlinear systems of equations

Practice Problems

1. Solve the following system of linear equations by substitution:

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

2. Solve the following system of linear equations by substitution:

{ \begin{array}{l} 3x - 2y = 12 \\ y = 4 \end{array} \}

3. Solve the following system of linear equations by substitution:

{ \begin{array}{l} 2x + 3y = 12 \\ y = 2 \end{array} \}

Glossary

• System of linear equations: A set of two or more linear equations that involve the same set of variables.
• Substitution method: A technique used to solve a system of linear equations by substituting the expression for one variable from one equation into the other equation.
• Linear equation: An equation in which the highest power of the variable(s) is 1.
• Variable: A symbol or expression that represents a value that can change.
  Solving a System of Linear Equations by Substitution: Q&A ===========================================================

Introduction

In our previous article, we discussed how to solve a system of linear equations by substitution. In this article, we will answer some frequently asked questions about solving a system of linear equations by substitution.

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of linear equations by substituting the expression for one variable from one equation into the other equation.

Q: When can I use the substitution method?

A: You can use the substitution method when one of the equations is already solved for one of the variables.

Q: How do I use the substitution method?

A: To use the substitution method, follow these steps:

1. Solve one of the equations for one of the variables.
2. Substitute the expression for the variable into the other equation.
3. Solve the resulting equation for the other variable.
4. Substitute the value of the second variable back into one of the original equations to find the value of the first variable.

Q: What if I have two equations with two variables, but neither equation is solved for one of the variables?

A: In this case, you can use the elimination method or the graphing method to solve the system of linear equations.

Q: Can I use the substitution method with systems of linear equations with three variables?

A: Yes, you can use the substitution method with systems of linear equations with three variables. However, it may be more complicated and require more steps.

Q: What if I get a contradiction when using the substitution method?

A: If you get a contradiction when using the substitution method, it means that the system of linear equations has no solution.

Q: Can I use the substitution method with nonlinear systems of equations?

A: No, the substitution method is only used with linear systems of equations. Nonlinear systems of equations require different methods, such as the elimination method or the graphing method.

Q: How do I know if the substitution method is the best method to use?

A: The substitution method is the best method to use when one of the equations is already solved for one of the variables. It is also a good method to use when the system of linear equations is simple and has only two variables.

Q: Can I use the substitution method with systems of linear equations with fractions?

A: Yes, you can use the substitution method with systems of linear equations with fractions. However, you may need to simplify the fractions before using the substitution method.

Q: What if I get a complex solution when using the substitution method?

A: If you get a complex solution when using the substitution method, it means that the system of linear equations has a complex solution.

Conclusion

In this article, we answered some frequently asked questions about solving a system of linear equations by substitution. The substitution method is a useful technique for solving systems of linear equations, especially when one of the equations is already solved for one of the variables.

Related Topics

• Solving a system of linear equations by elimination
• Solving a system of linear equations by graphing
• Systems of linear equations with three variables
• Nonlinear systems of equations

Practice Problems

1. Solve the following system of linear equations by substitution:

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

2. Solve the following system of linear equations by substitution:

{ \begin{array}{l} 3x - 2y = 12 \\ y = 4 \end{array} \}

3. Solve the following system of linear equations by substitution:

{ \begin{array}{l} 2x + 3y = 12 \\ y = 2 \end{array} \}

Glossary

• System of linear equations: A set of two or more linear equations that involve the same set of variables.
• Substitution method: A technique used to solve a system of linear equations by substituting the expression for one variable from one equation into the other equation.
• Linear equation: An equation in which the highest power of the variable(s) is 1.
• Variable: A symbol or expression that represents a value that can change.