What Are The Solutions To The Equation X − 7 X = 6 X-\frac{7}{x}=6 X − X 7 ​ = 6 ?A. X = − 7 X=-7 X = − 7 And X = 1 X=1 X = 1 B. X = − 6 X=-6 X = − 6 And X = − 1 X=-1 X = − 1 C. X = − 1 X=-1 X = − 1 And X = 7 X=7 X = 7 D. X = 1 X=1 X = 1 And X = 6 X=6 X = 6

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Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand the various techniques used to find the solutions. In this article, we will focus on solving a specific equation, $x-\frac{7}{x}=6$, and explore the different methods to find its solutions. We will also analyze the given options and determine which one is correct.

Understanding the Equation

The given equation is $x-\frac{7}{x}=6$. To solve this equation, we need to isolate the variable $x$. The first step is to get rid of the fraction by multiplying both sides of the equation by $x$. This will eliminate the fraction and make it easier to solve.

Multiplying Both Sides by $x$

Multiplying both sides of the equation by $x$ gives us:

$x^2 - 7 = 6x$

Rearranging the Equation

Now, we need to rearrange the equation to get all the terms on one side. We can do this by subtracting $6x$ from both sides of the equation:

$x^2 - 6x - 7 = 0$

Solving the Quadratic Equation

The equation $x^2 - 6x - 7 = 0$ is a quadratic equation, and we can solve it using the quadratic formula or factoring. In this case, we will use factoring to find the solutions.

Factoring the Quadratic Equation

To factor the quadratic equation, we need to find two numbers whose product is $-7$ and whose sum is $-6$. These numbers are $-7$ and $1$, so we can write the equation as:

$(x - 7)(x + 1) = 0$

Finding the Solutions

Now, we can find the solutions by setting each factor equal to zero and solving for $x$. This gives us:

$x - 7 = 0 \Rightarrow x = 7$

$x + 1 = 0 \Rightarrow x = -1$

Analyzing the Options

Now that we have found the solutions, we can analyze the given options to determine which one is correct. The options are:

A. $x=-7$ and $x=1$ B. $x=-6$ and $x=-1$ C. $x=-1$ and $x=7$ D. $x=1$ and $x=6$

Conclusion

Based on our analysis, we can see that the correct solution is option C, which states that the solutions to the equation $x-\frac{7}{x}=6$ are $x=-1$ and $x=7$.

Final Answer

The final answer is option C, which states that the solutions to the equation $x-\frac{7}{x}=6$ are $x=-1$ and $x=7$.

Additional Tips and Tricks

• When solving equations, it's essential to get rid of fractions by multiplying both sides by the denominator.
• Factoring is a powerful technique for solving quadratic equations, but it may not always be possible.
• When analyzing options, make sure to check each solution carefully to ensure that it satisfies the original equation.

Frequently Asked Questions

• Q: How do I solve a quadratic equation? A: You can solve a quadratic equation using the quadratic formula or factoring.
• Q: What is the quadratic formula? A: The quadratic formula is a mathematical formula that gives the solutions to a quadratic equation.
• Q: How do I factor a quadratic equation? A: To factor a quadratic equation, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Related Topics

• Solving linear equations
• Solving quadratic equations
• Factoring quadratic equations
• Quadratic formula

Conclusion

Solving equations is a fundamental concept in mathematics, and it's essential to understand the various techniques used to find the solutions. In this article, we have explored the different methods to solve the equation $x-\frac{7}{x}=6$ and analyzed the given options to determine which one is correct. We have also provided additional tips and tricks, as well as frequently asked questions, to help readers better understand the topic.

Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand the various techniques used to find the solutions. In this article, we will provide a comprehensive Q&A section to help readers better understand the topic. We will cover a range of questions, from basic concepts to advanced techniques, to provide a thorough understanding of solving equations.

Q: What is an equation?

A: An equation is a statement that two mathematical expressions are equal. It consists of two sides, a left-hand side and a right-hand side, separated by an equal sign (=).

Q: What is the difference between an equation and an expression?

A: An expression is a mathematical statement that does not contain an equal sign, while an equation is a statement that contains an equal sign.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable by performing operations on both sides of the equation. This can involve adding, subtracting, multiplying, or dividing both sides by the same value.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula or factoring. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that gives the solutions to a quadratic equation.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a system of equations?

A: To solve a system of equations, you need to find the values of the variables that satisfy all the equations in the system. This can involve using substitution, elimination, or graphing methods.

Q: What is the difference between a system of linear equations and a system of quadratic equations?

A: A system of linear equations is a set of linear equations that are solved simultaneously, while a system of quadratic equations is a set of quadratic equations that are solved simultaneously.

Q: How do I graph an equation?

A: To graph an equation, you need to plot the points that satisfy the equation on a coordinate plane. This can involve using a graphing calculator or plotting points by hand.

Q: What is the difference between a function and an equation?

A: A function is a relation between a set of inputs and a set of possible outputs, while an equation is a statement that two mathematical expressions are equal.

Q: How do I determine if an equation is a function?

A: To determine if an equation is a function, you need to check if each input corresponds to exactly one output.

Q: What is the difference between a linear function and a quadratic function?

A: A linear function is a function in which the highest power of the variable is 1, while a quadratic function is a function in which the highest power of the variable is 2.

Q: How do I graph a linear function?

A: To graph a linear function, you need to plot the points that satisfy the function on a coordinate plane. This can involve using a graphing calculator or plotting points by hand.

Q: What is the difference between a quadratic function and a polynomial function?

A: A quadratic function is a function in which the highest power of the variable is 2, while a polynomial function is a function in which the highest power of the variable is n, where n is a positive integer.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to plot the points that satisfy the function on a coordinate plane. This can involve using a graphing calculator or plotting points by hand.

Conclusion

Solving equations is a fundamental concept in mathematics, and it's essential to understand the various techniques used to find the solutions. In this article, we have provided a comprehensive Q&A section to help readers better understand the topic. We have covered a range of questions, from basic concepts to advanced techniques, to provide a thorough understanding of solving equations.

Additional Tips and Tricks

• When solving equations, it's essential to get rid of fractions by multiplying both sides by the denominator.
• Factoring is a powerful technique for solving quadratic equations, but it may not always be possible.
• When analyzing options, make sure to check each solution carefully to ensure that it satisfies the original equation.

Related Topics

• Solving linear equations
• Solving quadratic equations
• Factoring quadratic equations
• Quadratic formula
• Graphing equations
• Functions
• Linear functions
• Quadratic functions
• Polynomial functions

Final Answer

The final answer is that solving equations is a fundamental concept in mathematics, and it's essential to understand the various techniques used to find the solutions. By following the tips and tricks provided in this article, readers can gain a thorough understanding of solving equations and become proficient in solving a wide range of equations.