Consider The Following Equation:$\[ 7x - 9y^2 = 8 \\]1. Find Two Distinct Values Of \[$ Y \$\] That Satisfy The Equation For The Same Value Of \[$ X \$\]. Enter Your Answers As A Comma-separated List. If No Such Values Exist,

Solving the Equation: Finding Distinct Values of y for the Same Value of x

In this article, we will explore the given equation $7x - 9y^2 = 8$ and find two distinct values of $y$ that satisfy the equation for the same value of $x$. This problem involves solving a quadratic equation and understanding the relationship between the variables $x$ and $y$.

Understanding the Equation

The given equation is a quadratic equation in terms of $y$, where $x$ is a constant. The equation can be rewritten as $-9y^2 + 7x = 8$. To find the values of $y$ that satisfy the equation, we need to isolate $y$ and solve for its possible values.

Isolating y

To isolate $y$, we can rearrange the equation to get $-9y^2 = -7x + 8$. Dividing both sides by $-9$, we get $y^2 = \frac{7x - 8}{9}$.

Finding the Values of y

Now that we have isolated $y$, we can find its possible values by taking the square root of both sides of the equation. This gives us $y = \pm \sqrt{\frac{7x - 8}{9}}$.

Finding Two Distinct Values of y for the Same Value of x

To find two distinct values of $y$ for the same value of $x$, we need to find a value of $x$ that satisfies the equation and has two distinct solutions for $y$. Let's choose a value of $x$ that makes the expression inside the square root positive and equal to a perfect square.

Choosing a Value of x

Let's choose $x = 3$. Substituting this value into the equation, we get $y^2 = \frac{7(3) - 8}{9} = \frac{21 - 8}{9} = \frac{13}{9}$. This is not a perfect square, so we need to choose a different value of $x$.

Choosing Another Value of x

Let's choose $x = 4$. Substituting this value into the equation, we get $y^2 = \frac{7(4) - 8}{9} = \frac{28 - 8}{9} = \frac{20}{9}$. This is also not a perfect square, so we need to choose another value of $x$.

Choosing a Value of x that Makes the Expression Inside the Square Root a Perfect Square

Let's choose $x = 5$. Substituting this value into the equation, we get $y^2 = \frac{7(5) - 8}{9} = \frac{35 - 8}{9} = \frac{27}{9} = 3$. This is a perfect square, so we have found a value of $x$ that satisfies the equation and has two distinct solutions for $y$.

Finding the Values of y for x = 5

Now that we have found a value of $x$ that satisfies the equation and has two distinct solutions for $y$, we can find the values of $y$ by taking the square root of both sides of the equation. This gives us $y = \pm \sqrt{3}$.

Conclusion

In this article, we have explored the given equation $7x - 9y^2 = 8$ and found two distinct values of $y$ that satisfy the equation for the same value of $x$. We have chosen a value of $x$ that makes the expression inside the square root a perfect square and found the values of $y$ by taking the square root of both sides of the equation. The two distinct values of $y$ are $\sqrt{3}$ and $-\sqrt{3}$.

Final Answer

In our previous article, we explored the equation $7x - 9y^2 = 8$ and found two distinct values of $y$ that satisfy the equation for the same value of $x$. In this article, we will answer some frequently asked questions about the equation and provide additional insights into solving quadratic equations.

Q: What is the main difference between this equation and a standard quadratic equation? A: The main difference between this equation and a standard quadratic equation is that this equation has a variable $x$ that is not squared. In a standard quadratic equation, both variables are squared.

Q: How do I choose a value of $x$ that makes the expression inside the square root a perfect square? A: To choose a value of $x$ that makes the expression inside the square root a perfect square, you need to find a value of $x$ that makes the numerator of the expression a perfect square. You can do this by trial and error or by using algebraic manipulations.

Q: What if I choose a value of $x$ that does not make the expression inside the square root a perfect square? A: If you choose a value of $x$ that does not make the expression inside the square root a perfect square, you will not be able to find two distinct values of $y$ that satisfy the equation for the same value of $x$. In this case, you need to choose a different value of $x$.

Q: Can I use the quadratic formula to solve this equation? A: Yes, you can use the quadratic formula to solve this equation. The quadratic formula is $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = -9$, $b = 0$, and $c = 7x - 8$. Plugging these values into the quadratic formula, you get $y = \frac{0 \pm \sqrt{0 - 4(-9)(7x - 8)}}{2(-9)} = \frac{0 \pm \sqrt{252 - 288x}}{-18}$.

Q: How do I simplify the expression inside the square root? A: To simplify the expression inside the square root, you need to find a value of $x$ that makes the expression a perfect square. You can do this by trial and error or by using algebraic manipulations.

Q: Can I use a calculator to solve this equation? A: Yes, you can use a calculator to solve this equation. However, you need to be careful when using a calculator to solve quadratic equations, as it may not always give you the correct solution.

Conclusion

In this article, we have answered some frequently asked questions about the equation $7x - 9y^2 = 8$ and provided additional insights into solving quadratic equations. We have also discussed how to choose a value of $x$ that makes the expression inside the square root a perfect square and how to simplify the expression inside the square root.

Final Tips

• When solving quadratic equations, make sure to choose a value of $x$ that makes the expression inside the square root a perfect square.
• Use algebraic manipulations to simplify the expression inside the square root.
• Be careful when using a calculator to solve quadratic equations, as it may not always give you the correct solution.

Additional Resources

• For more information on solving quadratic equations, see our previous article on the topic.
• For more information on algebraic manipulations, see our article on algebraic manipulations.
• For more information on using calculators to solve quadratic equations, see our article on using calculators to solve quadratic equations.