Substitute the solution into the first equation

Oct 13, 2025 by talynt.org 52 views

Introduction

A system of equations is a set of two or more equations that contain the same variables. In this article, we will explore how to solve a system of equations using the given solution. We will analyze the system of equations and determine the correct steps that were taken to arrive at the solution.

System A: The Given Equations

The system of equations is given as:

{ \begin{array}{l} -x - 2y = 7 \\ 5x - 6y = -3 \\ \text{Solution: } (-3, -2) \end{array} \}

Step 1: Understanding the Solution

The solution to the system of equations is given as (-3, -2). This means that the values of x and y that satisfy both equations are x = -3 and y = -2.

Step 2: Substituting the Solution into the First Equation

To verify the solution, we can substitute the values of x and y into the first equation:

-x - 2y = 7

Substituting x = -3 and y = -2, we get:

-(-3) - 2(-2) = 7

Simplifying the equation, we get:

3 + 4 = 7

This confirms that the solution satisfies the first equation.

Step 3: Substituting the Solution into the Second Equation

Next, we can substitute the values of x and y into the second equation:

5x - 6y = -3

Substituting x = -3 and y = -2, we get:

5(-3) - 6(-2) = -3

Simplifying the equation, we get:

-15 + 12 = -3

This confirms that the solution satisfies the second equation.

Conclusion

Based on the analysis, it appears that the correct steps to solve the system of equations were:

Substitute the solution into the first equation: Verify that the solution satisfies the first equation.
Substitute the solution into the second equation: Verify that the solution satisfies the second equation.

By following these steps, we can confirm that the solution (-3, -2) is indeed the correct solution to the system of equations.

Why These Steps Were Necessary

The steps taken to solve the system of equations were necessary because they allowed us to verify that the solution satisfies both equations. By substituting the solution into each equation, we can confirm that the solution is indeed the correct solution to the system of equations.

What This Means for Solving Systems of Equations

When solving systems of equations, it is essential to verify that the solution satisfies each equation. By following the steps outlined above, we can ensure that the solution is correct and that we have a thorough understanding of the system of equations.

Common Mistakes to Avoid

When solving systems of equations, there are several common mistakes to avoid:

Not verifying the solution: Failing to verify that the solution satisfies each equation can lead to incorrect solutions.
Not following the correct steps: Failing to follow the correct steps can lead to incorrect solutions and a lack of understanding of the system of equations.

Best Practices for Solving Systems of Equations

To ensure that you are solving systems of equations correctly, follow these best practices:

Verify the solution: Always verify that the solution satisfies each equation.
Follow the correct steps: Always follow the correct steps to solve the system of equations.
Understand the system of equations: Always take the time to understand the system of equations and the solution.

By following these best practices, you can ensure that you are solving systems of equations correctly and that you have a thorough understanding of the system of equations.

Conclusion

Q: What is a system of equations?

A system of equations is a set of two or more equations that contain the same variables. In other words, it is a collection of equations that must be solved simultaneously to find the values of the variables.

Q: How do I solve a system of equations?

To solve a system of equations, you can use various methods such as substitution, elimination, or graphing. The method you choose will depend on the type of equations and the number of variables.

Q: What is the substitution method?

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one equation is easily solvable for one variable.

Q: What is the elimination method?

The elimination method involves adding or subtracting the equations to eliminate one variable. This method is useful when the coefficients of one variable are the same in both equations.

Q: What is the graphing method?

The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful when the equations are linear and the solution is a single point.

Q: How do I verify the solution?

To verify the solution, substitute the values of the variables into each equation and check if the equation is true. If the equation is true, then the solution is correct.

Q: What are some common mistakes to avoid when solving systems of equations?

Some common mistakes to avoid when solving systems of equations include:

Not verifying the solution
Not following the correct steps
Not understanding the system of equations
Not checking for extraneous solutions

Q: How do I check for extraneous solutions?

To check for extraneous solutions, substitute the values of the variables into each equation and check if the equation is true. If the equation is not true, then the solution is extraneous.

Q: What is an extraneous solution?

An extraneous solution is a solution that is not valid for the system of equations. It is a solution that satisfies one equation but not the other.

Q: How do I avoid extraneous solutions?

To avoid extraneous solutions, make sure to verify the solution by substituting the values of the variables into each equation. Also, make sure to check for extraneous solutions by substituting the values of the variables into each equation.

Q: What are some real-world applications of systems of equations?

Some real-world applications of systems of equations include:

Physics: Systems of equations are used to describe the motion of objects and the forces acting on them.
Engineering: Systems of equations are used to design and optimize systems such as electrical circuits and mechanical systems.
Economics: Systems of equations are used to model economic systems and make predictions about economic trends.
Computer Science: Systems of equations are used to solve problems in computer science such as graph theory and network flow.

Q: How do I use systems of equations in real-world applications?

To use systems of equations in real-world applications, you need to:

Identify the variables and the equations that describe the problem
Choose the appropriate method to solve the system of equations
Verify the solution by substituting the values of the variables into each equation
Check for extraneous solutions by substituting the values of the variables into each equation

By following these steps, you can use systems of equations to solve real-world problems and make predictions about the behavior of complex systems.