Solve The System Of Equations. (If There Is No Solution, Enter NO SOLUTION.)${ \begin{array}{c} \begin{cases} y = X^2 - X \ y = 8x - 18 \end{cases} \end{array} }$(x, Y) = ( ) (smaller X-value)(x, Y) = ( ) (larger X-value)

Mar 12, 2025 by ADMIN 223 views

Solve the System of Equations

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Introduction

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

$y = x^2 - x$
$y = 8x - 18$

Our goal is to find the values of $x$ and $y$ that satisfy both equations. We will use algebraic methods to solve this system of equations.

Step 1: Set the Equations Equal to Each Other

To solve the system of equations, 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.

x^2 - x = 8x - 18

Step 2: Simplify the Equation

Now, we need to simplify the equation by combining like terms.

x^2 - x - 8x + 18 = 0

x^2 - 9x + 18 = 0

Step 3: Factor the Quadratic Equation

The equation is a quadratic equation, and we can factor it as follows:

(x - 3)(x - 6) = 0

Step 4: Solve for x

Now, we can solve for $x$ by setting each factor equal to zero.

x - 3 = 0 \Rightarrow x = 3

x - 6 = 0 \Rightarrow x = 6

Step 5: Find the Corresponding y-Values

Now that we have the values of $x$ , we can find the corresponding $y$ -values by substituting $x$ into one of the original equations.

For $x = 3$ :

y = 3^2 - 3 = 9 - 3 = 6

For $x = 6$ :

y = 6^2 - 6 = 36 - 6 = 30

Step 6: Write the Solutions

The solutions to the system of equations are:

$(x, y) = (3, 6)$ (smaller $x$ -value)
$(x, y) = (6, 30)$ (larger $x$ -value)

Conclusion

In this article, we solved a system of equations using algebraic methods. We set the two equations equal to each other, simplified the equation, factored the quadratic equation, solved for $x$ , and found the corresponding $y$ -values. The solutions to the system of equations are $(3, 6)$ and $(6, 30)$ .

Discussion

Solving systems of equations is an important topic in mathematics, and it has many real-world applications. For example, in physics, we can use systems of equations to model the motion of objects. In economics, we can use systems of equations to model the behavior of markets.

Tips and Tricks

When solving systems of equations, it's often helpful to use algebraic methods such as substitution and elimination. We can also use graphing methods to visualize the solutions to the system of equations.

Common Mistakes

When solving systems of equations, it's easy to make mistakes such as:

Not setting the equations equal to each other
Not simplifying the equation
Not factoring the quadratic equation
Not solving for $x$ correctly
Not finding the corresponding $y$ -values

Real-World Applications

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

Modeling the motion of objects in physics
Modeling the behavior of markets in economics
Solving optimization problems in engineering
Solving problems in computer science

Final Thoughts

Solving systems of equations is an important topic in mathematics, and it has many real-world applications. By using algebraic methods such as substitution and elimination, we can solve systems of equations and find the solutions. With practice and experience, we can become proficient in solving systems of equations and apply this knowledge to real-world problems.

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Introduction

In our previous article, we solved a system of equations using algebraic methods. In this article, we will answer some frequently asked questions (FAQs) about solving systems of equations.

Q: What is a system of equations?

A system of equations is a set of two or more equations that involve the same variables. In this case, we had two equations:

$y = x^2 - x$
$y = 8x - 18$

Q: Why do we need to solve systems of equations?

We need to solve systems of equations to find the values of the variables that satisfy all the equations in the system. This is useful in many real-world applications, such as modeling the motion of objects in physics, modeling the behavior of markets in economics, and solving optimization problems in engineering.

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

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

Substitution method: We can substitute one equation into the other equation to solve for the variables.
Elimination method: We can add or subtract the equations to eliminate one of the variables.
Graphing method: We can graph the equations on a coordinate plane to find the intersection points.
Algebraic method: We can use algebraic techniques such as factoring and solving quadratic equations to solve the system.

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 setting the equations equal to each other
Not simplifying the equation
Not factoring the quadratic equation
Not solving for x correctly
Not finding the corresponding y-values

Q: How do I choose the right method for solving a system of equations?

The choice of method depends on the type of system and the variables involved. For example, if the system has two linear equations, the elimination method may be the best choice. If the system has a quadratic equation, the algebraic method may be the best choice.

Q: Can I use technology to solve systems of equations?

Yes, technology can be a powerful tool for solving systems of equations. Many graphing calculators and computer algebra systems (CAS) can solve systems of equations quickly and accurately.

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

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

Modeling the motion of objects in physics
Modeling the behavior of markets in economics
Solving optimization problems in engineering
Solving problems in computer science

Q: How can I practice solving systems of equations?

There are many resources available for practicing solving systems of equations, including:

Online practice problems
Textbooks and workbooks
Graphing calculators and CAS
Online courses and tutorials