Choosing the Correct Result

Oct 13, 2025 by miodeya.com 100 views

Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two equations with two variables, x and y. We will use a step-by-step approach to add the equations and then choose the correct result from the given options.

The System of Equations

The given system of equations is:

\begin{align*} x + y &= 8 \\ 2x^2 - y &= -5 \end{align*}

Adding the Equations

To add the equations, we need to combine like terms. We can start by adding the two equations together:

\begin{align*} x + y + 2x^2 - y &= 8 + (-5) \\ 2x^2 + x &= 3 \end{align*}

Choosing the Correct Result

Now that we have added the equations, we need to choose the correct result from the given options. Let's examine each option carefully:

Option A: $3x^2 = -13$
Option B: $3x^2 = 3$
Option C: $2x^2 + x = 3$
Option D: $2x^2 = -13$

Based on our calculations, we can see that the correct result is:

$2x^2 + x = 3$

This is the result we obtained when we added the two equations together.

Conclusion

Solving a system of equations involves finding the values of variables that satisfy multiple equations simultaneously. By adding the equations and choosing the correct result, we can solve the system of equations. In this article, we used a step-by-step approach to solve the given system of equations and obtained the correct result.

Tips and Tricks

Here are some tips and tricks to help you solve systems of equations:

Use the correct method: Make sure to use the correct method to add the equations, such as combining like terms.
Check your work: Always check your work to ensure that you have obtained the correct result.
Use algebraic manipulations: Algebraic manipulations, such as factoring and expanding, can help you solve systems of equations.

Common Mistakes

Here are some common mistakes to avoid when solving systems of equations:

Incorrect addition: Make sure to add the equations correctly, such as combining like terms.
Incorrect simplification: Make sure to simplify the equations correctly, such as canceling out common factors.
Incorrect solution: Make sure to obtain the correct solution, such as the values of the variables.

Real-World Applications

Solving systems of equations has many real-world applications, such as:

Physics: Solving systems of equations is used to model physical systems, such as the motion of objects.
Engineering: Solving systems of equations is used to design and optimize systems, such as electrical circuits.
Economics: Solving systems of equations is used to model economic systems, such as supply and demand.

Conclusion

Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will provide a Q&A section to help you understand the concept of solving systems of equations better.

Q: What is a system of equations?

A system of equations is a set of two or more equations that involve two or more variables. The goal is to find the values of the variables that satisfy all the equations simultaneously.

Q: How do I solve a system of equations?

To solve a system of equations, you can use various methods, such as:

Substitution method: Substitute one equation into the other equation to solve for one variable.
Elimination method: Add or subtract the equations to eliminate one variable.
Graphical method: Plot the equations on a graph to find the intersection point.

Q: What is the substitution method?

The substitution method involves substituting one equation into the other equation to solve for one variable. For example, if we have two equations:

\begin{align*} x + y &= 8 \\ 2x^2 - y &= -5 \end{align*}

We can substitute the first equation into the second equation to get:

2x^2 - (8 - x) = -5

Simplifying the equation, we get:

2x^2 - 8 + x = -5

Combine like terms:

2x^2 + x - 13 = 0

Q: What is the elimination method?

The elimination method involves adding or subtracting the equations to eliminate one variable. For example, if we have two equations:

\begin{align*} x + y &= 8 \\ 2x^2 - y &= -5 \end{align*}

We can add the two equations to eliminate the variable y:

x + y + 2x^2 - y = 8 + (-5)

Simplifying the equation, we get:

2x^2 + x = 3

Q: What is the graphical method?

The graphical method involves plotting the equations on a graph to find the intersection point. For example, if we have two equations:

\begin{align*} x + y &= 8 \\ 2x^2 - y &= -5 \end{align*}

We can plot the two equations on a graph to find the intersection point.

Q: How do I choose the correct method?

To choose the correct method, you need to consider the following factors:

Number of equations: If you have two equations, you can use the substitution or elimination method. If you have more than two equations, you may need to use the graphical method.
Number of variables: If you have two variables, you can use the substitution or elimination method. If you have more than two variables, you may need to use the graphical method.
Complexity of the equations: If the equations are simple, you can use the substitution or elimination method. If the equations are complex, you may need to use the graphical method.

Q: What are some common mistakes to avoid?

Here are some common mistakes to avoid when solving systems of equations:

Incorrect addition: Make sure to add the equations correctly, such as combining like terms.
Incorrect simplification: Make sure to simplify the equations correctly, such as canceling out common factors.
Incorrect solution: Make sure to obtain the correct solution, such as the values of the variables.

Conclusion

Solving a system of equations involves finding the values of variables that satisfy multiple equations simultaneously. By using the substitution, elimination, or graphical method, you can solve the system of equations. In this article, we provided a Q&A section to help you understand the concept of solving systems of equations better. We also provided tips and tricks to help you solve systems of equations and common mistakes to avoid.