Solve For { X $} . . . { \sqrt{1-7x} = 8 \} Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice:A. The Solution(s) Is/are { \square$}$.(Type An Integer Or A Simplified Fraction.

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Introduction

Solving equations involving square roots can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will focus on solving the equation $\sqrt{1-7x} = 8$ and provide step-by-step solutions to find the correct answer.

Understanding the Equation

The given equation is $\sqrt{1-7x} = 8$. To solve this equation, we need to isolate the variable $x$. The first step is to square both sides of the equation to eliminate the square root.

Squaring Both Sides

When we square both sides of the equation, we get:

$$\left(\sqrt{1-7x}\right)^2 = 8^2$$

This simplifies to:

$$1-7x = 64$$

Isolating the Variable

Now, we need to isolate the variable $x$. To do this, we can subtract 1 from both sides of the equation:

$$-7x = 63$$

Solving for $x$

Next, we can divide both sides of the equation by -7 to solve for $x$:

$$x = -\frac{63}{7}$$

Simplifying the Fraction

The fraction $-\frac{63}{7}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 7:

$$x = -\frac{9}{1}$$

Conclusion

Therefore, the solution to the equation $\sqrt{1-7x} = 8$ is $x = -9$. This is the correct answer to the given problem.

Discussion

Solving equations involving square roots requires careful manipulation of the equation to isolate the variable. In this case, we squared both sides of the equation to eliminate the square root and then isolated the variable $x$ by subtracting 1 and dividing by -7. The final answer is $x = -9$.

Final Answer

The final answer is: $\boxed{-9}$

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Introduction

Solving equations involving square roots can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will focus on solving the equation $\sqrt{1-7x} = 8$ and provide step-by-step solutions to find the correct answer.

Understanding the Equation

The given equation is $\sqrt{1-7x} = 8$. To solve this equation, we need to isolate the variable $x$. The first step is to square both sides of the equation to eliminate the square root.

Squaring Both Sides

When we square both sides of the equation, we get:

$$\left(\sqrt{1-7x}\right)^2 = 8^2$$

This simplifies to:

$$1-7x = 64$$

Isolating the Variable

Now, we need to isolate the variable $x$. To do this, we can subtract 1 from both sides of the equation:

$$-7x = 63$$

Solving for $x$

Next, we can divide both sides of the equation by -7 to solve for $x$:

$$x = -\frac{63}{7}$$

Simplifying the Fraction

The fraction $-\frac{63}{7}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 7:

$$x = -\frac{9}{1}$$

Conclusion

Therefore, the solution to the equation $\sqrt{1-7x} = 8$ is $x = -9$. This is the correct answer to the given problem.

Discussion

Solving equations involving square roots requires careful manipulation of the equation to isolate the variable. In this case, we squared both sides of the equation to eliminate the square root and then isolated the variable $x$ by subtracting 1 and dividing by -7. The final answer is $x = -9$.

Final Answer

The final answer is: $\boxed{-9}$

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Q: What is the first step in solving the equation $\sqrt{1-7x} = 8$?

A: The first step in solving the equation $\sqrt{1-7x} = 8$ is to square both sides of the equation to eliminate the square root.

Q: How do we simplify the equation after squaring both sides?

A: After squaring both sides of the equation, we get $1-7x = 64$. We can simplify this equation by subtracting 1 from both sides, which gives us $-7x = 63$.

Q: How do we solve for $x$ in the equation $-7x = 63$?

A: To solve for $x$ in the equation $-7x = 63$, we can divide both sides of the equation by -7, which gives us $x = -\frac{63}{7}$.

Q: Can we simplify the fraction $-\frac{63}{7}$?

A: Yes, we can simplify the fraction $-\frac{63}{7}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 7. This gives us $x = -\frac{9}{1}$.

Q: What is the final answer to the equation $\sqrt{1-7x} = 8$?

A: The final answer to the equation $\sqrt{1-7x} = 8$ is $x = -9$.

Q: What is the most important thing to remember when solving equations involving square roots?

A: The most important thing to remember when solving equations involving square roots is to carefully manipulate the equation to isolate the variable. In this case, we squared both sides of the equation to eliminate the square root and then isolated the variable $x$ by subtracting 1 and dividing by -7.

Q: Can you provide an example of a similar equation that can be solved using the same steps?

A: Yes, an example of a similar equation that can be solved using the same steps is $\sqrt{4-12x} = 5$. We can follow the same steps to solve for $x$ in this equation.

Q: How do we know if the solution to the equation is valid?

A: We can check if the solution to the equation is valid by plugging it back into the original equation. If the solution satisfies the original equation, then it is a valid solution.

Q: What if the solution to the equation is not a real number?

A: If the solution to the equation is not a real number, then it is not a valid solution. In this case, we need to re-examine our steps and make sure that we did not make any mistakes.

Q: Can you provide a summary of the steps to solve the equation $\sqrt{1-7x} = 8$?

A: Yes, the steps to solve the equation $\sqrt{1-7x} = 8$ are:

1. Square both sides of the equation to eliminate the square root.
2. Simplify the equation by subtracting 1 from both sides.
3. Solve for $x$ by dividing both sides of the equation by -7.
4. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor.
5. Check if the solution is valid by plugging it back into the original equation.