To Solve The System Of Equations Below, Zach Isolated X 2 X^2 X 2 In The First Equation And Then Substituted It Into The Second Equation. What Was The Resulting Equation?$[ \begin{cases} x^2 + Y^2 = 25 \ x^2 - \frac{y^2}{9} = 1

Introduction

Solving systems 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 explore a step-by-step approach to solving systems of equations, using the example of Zach, who isolated $x^2$ in the first equation and then substituted it into the second equation.

The System of Equations

The system of equations provided by Zach is:

$\[ \begin{cases} x^2 + y^2 = 25 \\ x^2 - \frac{y^2}{9} = 1 \end{cases}$

Isolating $x^2$ in the First Equation

To isolate $x^2$ in the first equation, we can start by subtracting $y^2$ from both sides of the equation:

$x^2 = 25 - y^2$

This allows us to express $x^2$ in terms of $y^2$.

Substituting $x^2$ into the Second Equation

Now that we have isolated $x^2$ in the first equation, we can substitute it into the second equation:

$x^2 - \frac{y^2}{9} = 1$

Substituting $x^2 = 25 - y^2$ into the second equation, we get:

$(25 - y^2) - \frac{y^2}{9} = 1$

Simplifying the Equation

To simplify the equation, we can start by combining like terms:

$25 - \frac{10y^2}{9} = 1$

Next, we can subtract 25 from both sides of the equation:

$-\frac{10y^2}{9} = -24$

Multiplying Both Sides by -9

To eliminate the fraction, we can multiply both sides of the equation by -9:

$-10y^2 = -216$

Dividing Both Sides by -10

Finally, we can divide both sides of the equation by -10 to solve for $y^2$:

$y^2 = \frac{216}{10}$

$y^2 = 21.6$

Conclusion

In this article, we have walked through the step-by-step process of solving a system of equations by isolating $x^2$ in the first equation and then substituting it into the second equation. By following these steps, we were able to simplify the equation and solve for $y^2$. This approach can be applied to a wide range of systems of equations, making it a valuable tool for mathematicians and scientists alike.

Additional Resources

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

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

Frequently Asked Questions

Q: What is the first step in solving a system of equations? A: The first step is to isolate one of the variables in one of the equations.

Q: How do I substitute an expression into an equation? A: To substitute an expression into an equation, simply replace the variable with the expression.

Q: What is the final step in solving a system of equations? A: The final step is to solve for the remaining variable(s).

Glossary of Terms

• System of equations: A set of two or more equations that involve multiple variables.
• Isolate: To express a variable in terms of other variables.
• Substitute: To replace a variable with an expression.
• Simplify: To reduce an equation to its simplest form.
  Frequently Asked Questions: 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 multiple variables. Each equation is a statement that two or more expressions are equal, and the system is a collection of these statements.

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

Solving systems of equations is essential in many areas of mathematics and science, such as physics, engineering, economics, and computer science. It helps us to model real-world problems, make predictions, and understand complex relationships between variables.

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

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

• Substitution method: This involves isolating one variable in one equation and substituting it into the other equation.
• Elimination method: This involves adding or subtracting equations to eliminate one variable.
• Graphical method: This involves graphing the equations on a coordinate plane and finding the point of intersection.
• Matrix method: This involves using matrices to represent the system of equations and solve for the variables.

Q: What is the substitution method?

The substitution method involves isolating one variable in one equation and substituting it into the other equation. This method is useful when one of the equations is easily solvable for one variable.

Q: What is the elimination method?

The elimination method involves adding or subtracting 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 graphical method?

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

Q: What is the matrix method?

The matrix method involves using matrices to represent the system of equations and solve for the variables. This method is useful when the system of equations is large and complex.

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

The best method for solving a system of equations depends on the specific problem and the variables involved. You should choose the method that is most suitable for the problem and the variables.

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 checking the solution: Make sure to check the solution to ensure that it satisfies both equations.
• Not using the correct method: Choose the method that is most suitable for the problem and the variables.
• Not simplifying the equations: Simplify the equations before solving them to make the solution easier to find.

Q: How do I check the solution to a system of equations?

To check the solution to a system of equations, substitute the values of the variables into both equations and check if they are true.

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

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

• Physics: Solving systems of equations is used to model the motion of objects and predict their behavior.
• Engineering: Solving systems of equations is used to design and optimize systems, such as bridges and buildings.
• Economics: Solving systems of equations is used to model economic systems and make predictions about the behavior of markets.
• Computer Science: Solving systems of equations is used to model complex systems and make predictions about their behavior.

Q: How do I practice solving systems of equations?

To practice solving systems of equations, try the following:

• Work on practice problems: Practice solving systems of equations using different methods and variables.
• Use online resources: Use online resources, such as Khan Academy and Mathway, to practice solving systems of equations.
• Join a study group: Join a study group or find a study partner to practice solving systems of equations together.