Creating a Table of Solutions

Oct 13, 2025 by pynkocean.com 31 views

Understanding the Equation

The given equation is $\frac{x}{6} + 7 = -1$ . To determine whether a value of $x$ is a solution to this equation, we need to isolate the variable $x$ and find its value. The equation involves a fraction, so we will need to use algebraic manipulations to solve for $x$ .

Step 1: Subtract 7 from Both Sides

To isolate the fraction, we start by subtracting 7 from both sides of the equation:

\frac{x}{6} + 7 - 7 = -1 - 7

This simplifies to:

\frac{x}{6} = -8

Step 2: Multiply Both Sides by 6

To get rid of the fraction, we multiply both sides of the equation by 6:

6 \times \frac{x}{6} = -8 \times 6

This simplifies to:

x = -48

Is x a Solution to the Equation?

Now that we have found the value of $x$ , we can determine whether it is a solution to the original equation. We substitute $x = -48$ into the original equation:

\frac{-48}{6} + 7 = -1

Simplifying this expression, we get:

-8 + 7 = -1

Which is true, since $-1 = -1$ . Therefore, $x = -48$ is a solution to the equation.

Creating a Table of Solutions

To summarize our findings, we can create a table with the values of $x$ and their corresponding solutions:

$x$	Is it a solution?
-48	Yes

Conclusion

In this article, we have solved the equation $\frac{x}{6} + 7 = -1$ and determined whether a value of $x$ is a solution to the equation. We found that $x = -48$ is a solution to the equation. This value of $x$ satisfies the equation, and we can confirm this by substituting it into the original equation.

Discussion

The equation $\frac{x}{6} + 7 = -1$ is a simple linear equation that can be solved using basic algebraic manipulations. The solution to the equation is $x = -48$ , which can be verified by substituting it into the original equation. This equation is a good example of how to solve a linear equation with a fraction, and it can be used as a starting point for more complex equations.

Further Exploration

To further explore this topic, we can consider other values of $x$ and determine whether they are solutions to the equation. We can also consider more complex equations that involve fractions and other algebraic manipulations.

Real-World Applications

The equation $\frac{x}{6} + 7 = -1$ may not have any direct real-world applications, but it can be used as a starting point for more complex equations that have real-world applications. For example, we can use this equation to model a situation where a quantity is increased by a certain amount and then decreased by a certain amount.

Mathematical Concepts

The equation $\frac{x}{6} + 7 = -1$ involves several mathematical concepts, including:

Linear equations: The equation is a linear equation, which means that it can be written in the form $ax + b = c$ , where $a$ , $b$ , and $c$ are constants.
Fractions: The equation involves a fraction, which is a way of expressing a part of a whole.
Algebraic manipulations: The equation can be solved using basic algebraic manipulations, such as adding, subtracting, multiplying, and dividing.

Conclusion

In conclusion, the equation $\frac{x}{6} + 7 = -1$ is a simple linear equation that can be solved using basic algebraic manipulations. The solution to the equation is $x = -48$ , which can be verified by substituting it into the original equation. This equation is a good example of how to solve a linear equation with a fraction, and it can be used as a starting point for more complex equations.

Q: What is the solution to the equation $\frac{x}{6} + 7 = -1$ ?

A: The solution to the equation is $x = -48$ . This value of $x$ satisfies the equation, and it can be verified by substituting it into the original equation.

Q: How do I solve the equation $\frac{x}{6} + 7 = -1$ ?

A: To solve the equation, you can start by subtracting 7 from both sides of the equation. This will give you $\frac{x}{6} = -8$ . Then, you can multiply both sides of the equation by 6 to get rid of the fraction. This will give you $x = -48$ .

Q: What if I get a different value of $x$ when I solve the equation?

A: If you get a different value of $x$ when you solve the equation, it may be because you made a mistake in your calculations. Double-check your work to make sure that you are following the correct steps. If you are still having trouble, you can try using a different method to solve the equation.

Q: Can I use this equation to model a real-world situation?

A: While the equation $\frac{x}{6} + 7 = -1$ may not have any direct real-world applications, it can be used as a starting point for more complex equations that have real-world applications. For example, you can use this equation to model a situation where a quantity is increased by a certain amount and then decreased by a certain amount.

Q: What are some common mistakes to avoid when solving this equation?

A: Some common mistakes to avoid when solving this equation include:

Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when solving the equation.
Not simplifying the equation: Make sure to simplify the equation as much as possible before solving for $x$ .
Not checking your work: Make sure to check your work to make sure that you are getting the correct solution.

Q: Can I use this equation to solve for $x$ in other equations?

A: Yes, you can use this equation to solve for $x$ in other equations. For example, you can use this equation to solve for $x$ in the equation $\frac{x}{6} + 7 = 3$ . Simply substitute $x = -48$ into the equation and solve for $x$ .

Q: What are some other equations that involve fractions?

A: Some other equations that involve fractions include:

$\frac{x}{4} + 2 = 5$
$\frac{x}{3} - 1 = 2$
$\frac{x}{2} + 3 = 1$

These equations can be solved using the same methods as the equation $\frac{x}{6} + 7 = -1$ .

Q: Can I use this equation to solve for $x$ in equations with multiple variables?

A: Yes, you can use this equation to solve for $x$ in equations with multiple variables. For example, you can use this equation to solve for $x$ in the equation $\frac{x}{6} + 7 = -1 + 2y$ . Simply substitute $x = -48$ into the equation and solve for $y$ .

Q: What are some other mathematical concepts that involve fractions?

A: Some other mathematical concepts that involve fractions include:

Ratios: A ratio is a comparison of two numbers.
Proportions: A proportion is a statement that two ratios are equal.
Percentages: A percentage is a way of expressing a part of a whole as a fraction of 100.

These concepts can be used to solve problems that involve fractions, and they can be applied to a wide range of real-world situations.