Find The Ordered Pair Solutions For The System Of Equations:${ \begin{cases} y = X^2 - 2x + 3 \ y = -5x + 1 \end{cases} }$

Mar 9, 2025 by ADMIN 124 views

**Solving a System of Equations: Finding Ordered Pair Solutions**

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. In this article, we will focus on finding the ordered pair solutions for a system of two equations. The system of equations we will be working with is:

{ \begin{cases} y = x^2 - 2x + 3 \\ y = -5x + 1 \end{cases} \}

Understanding the System of Equations

The first equation is a quadratic equation in the form of $y = ax^2 + bx + c$ , where $a = 1$ , $b = -2$ , and $c = 3$ . The second equation is a linear equation in the form of $y = mx + b$ , where $m = -5$ and $b = 1$ .

Setting Up the System of Equations

To find the ordered pair solutions, we need to set the two equations equal to each other. This is because both equations are equal to $y$ , so we can set them equal to each other and solve for $x$ .

{ x^2 - 2x + 3 = -5x + 1 \}

Solving for x

To solve for $x$ , we need to isolate the variable $x$ on one side of the equation. We can do this by adding $5x$ to both sides of the equation and subtracting $3$ from both sides.

{ x^2 + 3x - 2 = 0 \}

Factoring the Quadratic Equation

The quadratic equation $x^2 + 3x - 2 = 0$ can be factored as $(x + 2)(x - 1) = 0$ . This means that either $(x + 2) = 0$ or $(x - 1) = 0$ .

Solving for x

Solving for $x$ , we get:

{ x + 2 = 0 \implies x = -2 \}

{ x - 1 = 0 \implies x = 1 \}

Finding the Ordered Pair Solutions

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 $y$ . We will use the first equation $y = x^2 - 2x + 3$ .

For $x = -2$ , we get:

{ y = (-2)^2 - 2(-2) + 3 \}

{ y = 4 + 4 + 3 \}

{ y = 11 \}

So, the ordered pair solution for $x = -2$ is $(-2, 11)$ .

For $x = 1$ , we get:

{ y = (1)^2 - 2(1) + 3 \}

{ y = 1 - 2 + 3 \}

{ y = 2 \}

So, the ordered pair solution for $x = 1$ is $(1, 2)$ .

Conclusion

In this article, we have found the ordered pair solutions for the system of equations $y = x^2 - 2x + 3$ and $y = -5x + 1$ . We have used the method of substitution to solve for $x$ and then found the corresponding values of $y$ . The ordered pair solutions are $(-2, 11)$ and $(1, 2)$ .

Applications of Systems of Equations

Systems of equations have many real-world applications, including:

Physics and Engineering: Systems of equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
Economics: Systems of equations are used to model economic systems, such as supply and demand curves.
Computer Science: Systems of equations are used in computer science to solve problems, such as linear programming and graph theory.

Tips for Solving Systems of Equations

Here are some tips for solving systems of equations:

Use the method of substitution: This involves substituting one equation into the other equation to solve for the variable.
Use the method of elimination: This involves adding or subtracting the equations to eliminate one of the variables.
Graph the equations: This involves graphing the equations on a coordinate plane to find the point of intersection.

Common Mistakes to Avoid

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

Not checking the solutions: Make sure to check the solutions to ensure that they are correct.
Not using the correct method: Make sure to use the correct method, such as substitution or elimination.
Not graphing the equations: Make sure to graph the equations to find the point of intersection.

Conclusion

In conclusion, solving systems of equations is an important skill in mathematics. By using the method of substitution or elimination, we can find the ordered pair solutions for a system of equations. With practice and patience, we can become proficient in solving systems of equations and apply them to real-world problems.

Introduction

Solving systems of equations can be a challenging task, but with the right guidance, it can become a breeze. In this article, we will answer some of the most frequently asked questions about solving systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

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

A: There are two main methods for solving systems of equations: the method of substitution and the method of elimination.

Q: What is the method of substitution?

A: The method of substitution involves substituting one equation into the other equation to solve for the variable.

Q: What is the method of elimination?

A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I choose which method to use?

A: You can choose which method to use based on the type of equations you are working with. If the equations are linear, you can use the method of elimination. If the equations are quadratic, you can use the method of substitution.

Q: What is the point of intersection?

A: The point of intersection is the point where the two equations meet. This is the solution to the system of equations.

Q: How do I find the point of intersection?

A: You can find the point of intersection by graphing the equations on a coordinate plane or by using the method of substitution or elimination.

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:

Not checking the solutions
Not using the correct method
Not graphing the equations

Q: How do I check my solutions?

A: You can check your solutions by plugging them back into the original equations to make sure they are true.

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

A: Systems of equations have many real-world applications, including:

Physics and engineering
Economics
Computer science

Q: How can 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.

Q: What are some tips for solving systems of equations?

A: Some tips for solving systems of equations include:

Use the method of substitution or elimination
Graph the equations on a coordinate plane
Check your solutions

Conclusion

Solving systems of equations can be a challenging task, but with the right guidance, it can become a breeze. By following the tips and techniques outlined in this article, you can become proficient in solving systems of equations and apply them to real-world problems.

Additional Resources

For more information on solving systems of equations, check out the following resources:

Khan Academy: Systems of Equations
Mathway: Systems of Equations
Wolfram Alpha: Systems of Equations

Practice Problems

Try solving the following systems of equations:

\begin{cases} y = x^2 - 2x + 3 \ y = -5x + 1 \end{cases} }$

\begin{cases} y = 2x - 3 \ y = x + 2 \end{cases} }$

\begin{cases} y = x^2 + 4x - 5 \ y = -2x + 3 \end{cases} }$

Answer Key

$(-2, 11)$ and $(1, 2)$
$(5, 7)$ and $(-3, -1)$
$(1, -2)$ and $(-3, 7)$