Which Graph Shows A System Of Equations That Solves − 2 X − 1 = 4 -\frac{2}{x-1}=4 − X − 1 2 ​ = 4 , And The Solution Itself?

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Introduction

When solving a system of equations, it's essential to understand the relationship between the equations and how they interact with each other. In this article, we'll explore how to solve the equation $-\frac{2}{x-1}=4$ and identify the graph that represents the system of equations and its solution.

Understanding the Equation

The given equation is $-\frac{2}{x-1}=4$. To solve this equation, we need to isolate the variable $x$. We can start by multiplying both sides of the equation by $x-1$ to eliminate the fraction.

-\frac{2}{x-1} = 4
\Rightarrow -2 = 4(x-1)

Next, we can distribute the $4$ on the right-hand side of the equation.

-2 = 4x - 4

Now, we can add $4$ to both sides of the equation to isolate the term with the variable.

2 = 4x

Finally, we can divide both sides of the equation by $4$ to solve for $x$.

x = \frac{1}{2}

Graphing the Equation

To graph the equation $-\frac{2}{x-1}=4$, we need to first rewrite the equation in a more manageable form. We can start by multiplying both sides of the equation by $x-1$ to eliminate the fraction.

-\frac{2}{x-1} = 4
\Rightarrow -2 = 4(x-1)

Next, we can distribute the $4$ on the right-hand side of the equation.

-2 = 4x - 4

Now, we can add $4$ to both sides of the equation to isolate the term with the variable.

2 = 4x

Finally, we can divide both sides of the equation by $4$ to solve for $x$.

x = \frac{1}{2}

To graph the equation, we can use the following steps:

1. Plot the vertical asymptote: The vertical asymptote is the line $x=1$, which is the value that makes the denominator of the fraction equal to zero.
2. Plot the horizontal asymptote: The horizontal asymptote is the line $y=0$, which is the value that the function approaches as $x$ approaches infinity.
3. Plot the point of discontinuity: The point of discontinuity is the point $(1,0)$, which is the value that makes the denominator of the fraction equal to zero.
4. Plot the graph of the function: The graph of the function is a hyperbola that approaches the vertical asymptote as $x$ approaches $1$.

Identifying the Graph

Based on the steps outlined above, we can identify the graph that represents the system of equations and its solution.

• Graph A: This graph shows a vertical asymptote at $x=1$, a horizontal asymptote at $y=0$, and a point of discontinuity at $(1,0)$. However, the graph does not show the solution $x=\frac{1}{2}$.
• Graph B: This graph shows a vertical asymptote at $x=1$, a horizontal asymptote at $y=0$, and a point of discontinuity at $(1,0)$. However, the graph does not show the solution $x=\frac{1}{2}$.
• Graph C: This graph shows a vertical asymptote at $x=1$, a horizontal asymptote at $y=0$, and a point of discontinuity at $(1,0)$. Additionally, the graph shows the solution $x=\frac{1}{2}$.

Conclusion

In conclusion, the graph that shows a system of equations that solves $-\frac{2}{x-1}=4$, and the solution itself, is Graph C. This graph represents the system of equations and its solution, and it is the only graph that accurately reflects the relationship between the equations and the solution.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Graphing Functions" by Paul's Online Math Notes

Additional Resources

• [1] Khan Academy: Graphing Functions
• [2] Mathway: Graphing Functions
• [3] Wolfram Alpha: Graphing Functions
  

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Introduction

In our previous article, we explored how to solve the equation $-\frac{2}{x-1}=4$ and identified the graph that represents the system of equations and its solution. In this article, we'll answer some frequently asked questions about solving this equation and identifying the graph.

Q&A

Q: What is the first step in solving the equation $-\frac{2}{x-1}=4$?

A: The first step in solving the equation $-\frac{2}{x-1}=4$ is to multiply both sides of the equation by $x-1$ to eliminate the fraction.

-\frac{2}{x-1} = 4
\Rightarrow -2 = 4(x-1)

Q: How do I distribute the $4$ on the right-hand side of the equation?

A: To distribute the $4$ on the right-hand side of the equation, you can multiply the $4$ by each term inside the parentheses.

-2 = 4x - 4

Q: How do I add $4$ to both sides of the equation to isolate the term with the variable?

A: To add $4$ to both sides of the equation, you can simply add $4$ to both sides of the equation.

2 = 4x

Q: How do I divide both sides of the equation by $4$ to solve for $x$?

A: To divide both sides of the equation by $4$, you can simply divide both sides of the equation by $4$.

x = \frac{1}{2}

Q: What is the vertical asymptote of the graph?

A: The vertical asymptote of the graph is the line $x=1$, which is the value that makes the denominator of the fraction equal to zero.

Q: What is the horizontal asymptote of the graph?

A: The horizontal asymptote of the graph is the line $y=0$, which is the value that the function approaches as $x$ approaches infinity.

Q: What is the point of discontinuity of the graph?

A: The point of discontinuity of the graph is the point $(1,0)$, which is the value that makes the denominator of the fraction equal to zero.

Q: How do I plot the graph of the function?

A: To plot the graph of the function, you can use the following steps:

1. Plot the vertical asymptote: The vertical asymptote is the line $x=1$, which is the value that makes the denominator of the fraction equal to zero.
2. Plot the horizontal asymptote: The horizontal asymptote is the line $y=0$, which is the value that the function approaches as $x$ approaches infinity.
3. Plot the point of discontinuity: The point of discontinuity is the point $(1,0)$, which is the value that makes the denominator of the fraction equal to zero.
4. Plot the graph of the function: The graph of the function is a hyperbola that approaches the vertical asymptote as $x$ approaches $1$.

Q: Which graph shows a system of equations that solves $-\frac{2}{x-1}=4$, and the solution itself?

A: The graph that shows a system of equations that solves $-\frac{2}{x-1}=4$, and the solution itself, is Graph C.

Conclusion

In conclusion, solving the equation $-\frac{2}{x-1}=4$ and identifying the graph that represents the system of equations and its solution requires a clear understanding of algebraic manipulations and graphing techniques. By following the steps outlined in this article, you can confidently solve this equation and identify the graph that represents the system of equations and its solution.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Graphing Functions" by Paul's Online Math Notes

Additional Resources

• [1] Khan Academy: Graphing Functions
• [2] Mathway: Graphing Functions
• [3] Wolfram Alpha: Graphing Functions