Solve The Following System Of Equations Algebraically:${ \begin{array}{l} y = X^2 + 7x - 10 \ y = X - 10 \end{array} }$

Mar 8, 2025 by ADMIN 121 views

Solving the System of Equations Algebraically

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Introduction

Solving a system of equations algebraically involves finding the values of the variables that satisfy all the equations in the system. In this case, we have two equations:

$y = x^2 + 7x - 10$
$y = x - 10$

Our goal is to find the values of $x$ and $y$ that satisfy both equations simultaneously.

Step 1: Equate the Two Equations

To solve the system of equations, we can start by equating the two equations. Since both equations are equal to $y$ , we can set them equal to each other:

$x^2 + 7x - 10 = x - 10$

Step 2: Simplify the Equation

Next, we can simplify the equation by combining like terms:

$x^2 + 7x - 10 - x + 10 = 0$

This simplifies to:

$x^2 + 6x = 0$

Step 3: Factor the Quadratic Equation

The quadratic equation $x^2 + 6x = 0$ can be factored as:

$x(x + 6) = 0$

Step 4: Solve for x

To solve for $x$ , we can set each factor equal to zero:

$x = 0$ or $x + 6 = 0$

Solving for $x$ in the second equation, we get:

$x = -6$

Step 5: Find the Corresponding Values of y

Now that we have found the values of $x$ , we can substitute them into one of the original equations to find the corresponding values of $y$ . Let's use the second equation:

$y = x - 10$

Substituting $x = 0$ , we get:

$y = 0 - 10$

$y = -10$

Substituting $x = -6$ , we get:

$y = -6 - 10$

$y = -16$

Step 6: Write the Final Solution

Therefore, the final solution to the system of equations is:

$(x, y) = (0, -10)$ or $(x, y) = (-6, -16)$

Conclusion

Solving a system of equations algebraically involves finding the values of the variables that satisfy all the equations in the system. In this case, we used the method of substitution to solve the system of equations. We equated the two equations, simplified the resulting equation, factored the quadratic equation, and solved for $x$ . Finally, we found the corresponding values of $y$ and wrote the final solution.

Key Takeaways

To solve a system of equations algebraically, we can use the method of substitution.
We can equate the two equations and simplify the resulting equation.
We can factor the quadratic equation and solve for $x$ .
We can find the corresponding values of $y$ by substituting the values of $x$ into one of the original equations.

Real-World Applications

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

Physics and Engineering: Solving systems of equations is used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
Computer Science: Solving systems of equations is used in computer graphics, game development, and machine learning.
Economics: Solving systems of equations is used to model economic systems, such as supply and demand curves.

Tips and Tricks

When solving a system of equations, it's often helpful to use the method of substitution.
We can use the method of elimination to solve a system of equations by adding or subtracting the equations.
We can use the method of graphing to solve a system of equations by graphing the equations on a coordinate plane.

Common Mistakes

One common mistake when solving a system of equations is to forget to check the solutions.
Another common mistake is to use the wrong method to solve the system of equations.
We should always check our solutions to make sure they satisfy both equations in the system.

Conclusion

Solving a system of equations algebraically involves finding the values of the variables that satisfy all the equations in the system. We used the method of substitution to solve the system of equations, equated the two equations, simplified the resulting equation, factored the quadratic equation, and solved for $x$ . Finally, we found the corresponding values of $y$ and wrote the final solution.

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Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other through the variables in the equations.

Q: How do I know if I have a system of equations?

A: You have a system of equations if you have two or more equations that are related to each other through the variables in the equations.

Q: What are the different methods for solving systems of equations?

A: There are several methods for solving systems of equations, including:

Substitution Method: This method involves substituting one equation into the other equation to solve for the variables.
Elimination Method: This method involves adding or subtracting the equations to eliminate one of the variables.
Graphing Method: This method involves graphing the equations on a coordinate plane to find the intersection point.

Q: What is the substitution method?

A: The substitution method involves substituting one equation into the other equation to solve for the variables. This method is useful when one of the equations is linear and the other equation is quadratic.

Q: What is the elimination method?

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

Q: What is the graphing method?

A: The graphing method involves graphing the equations on a coordinate plane to find the intersection point. This method is useful when the equations are linear.

Q: How do I choose the method to use?

A: You should choose the method that is most suitable for the type of equations you are working with. For example, if you have two linear equations, the graphing method may be the best choice.

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

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

Forgetting to check the solutions: Make sure to check the solutions to ensure that they satisfy both equations in the system.
Using the wrong method: Choose the method that is most suitable for the type of equations you are working with.
Not simplifying the equations: Make sure to simplify the equations before solving them.

Q: How do I check the solutions?

A: To check the solutions, substitute the values of the variables into both equations and make sure that they satisfy both equations.

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

A: Solving systems of equations has many real-world applications, including:

Physics and Engineering: Solving systems of equations is used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
Computer Science: Solving systems of equations is used in computer graphics, game development, and machine learning.
Economics: Solving systems of equations is used to model economic systems, such as supply and demand curves.

Q: How do I practice solving systems of equations?

A: You can practice solving systems of equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of equations on your own using the methods described above.

Q: What are some resources for learning more about solving systems of equations?

A: Some resources for learning more about solving systems of equations include:

Textbooks: There are many textbooks available that cover the topic of solving systems of equations.
Online Resources: There are many online resources available that provide examples and exercises for solving systems of equations.
Video Tutorials: There are many video tutorials available that provide step-by-step instructions for solving systems of equations.

Conclusion

Solving systems of equations is an important skill that has many real-world applications. By understanding the different methods for solving systems of equations, you can choose the method that is most suitable for the type of equations you are working with. Remember to check the solutions and simplify the equations before solving them. With practice and patience, you can become proficient in solving systems of equations.