Which System Of Equations Could Be Graphed To Solve The Equation Below? Log ⁡ 0.5 X = Log ⁡ 3 ( 2 + X \log_{0.5} X = \log_3 (2 + X Lo G 0.5 ​ X = Lo G 3 ​ ( 2 + X ]A. Y 1 = Log ⁡ 0.5 X , Y 2 = Log ⁡ 3 2 + X Y_1 = \frac{\log 0.5}{x}, \quad Y_2 = \frac{\log 3}{2 + X} Y 1 ​ = X L O G 0.5 ​ , Y 2 ​ = 2 + X L O G 3 ​ B. $y_1 = \frac{\log X}{\log 0.5}, \quad Y_2 = \frac{\log (2 +

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Which System of Equations Could Be Graphed to Solve the Equation Below?

When dealing with logarithmic equations, it can be challenging to find the solution directly. However, by using a system of equations and graphing, we can find the solution to the equation. In this article, we will explore which system of equations could be graphed to solve the equation log⁡0.5x=log⁡3(2+x)\log_{0.5} x = \log_3 (2 + x).

Before we dive into the systems of equations, let's understand the given equation. The equation is log⁡0.5x=log⁡3(2+x)\log_{0.5} x = \log_3 (2 + x). This equation involves logarithms with different bases, 0.5 and 3. To solve this equation, we need to find the value of x that satisfies both sides of the equation.

The first system of equations is:

y1=log⁡0.5x,y2=log⁡32+xy_1 = \frac{\log 0.5}{x}, \quad y_2 = \frac{\log 3}{2 + x}

To understand this system, let's break it down. The first equation is y1=log⁡0.5xy_1 = \frac{\log 0.5}{x}. This equation represents a logarithmic function with a base of 0.5 and a variable x. The second equation is y2=log⁡32+xy_2 = \frac{\log 3}{2 + x}. This equation represents a logarithmic function with a base of 3 and a variable x.

To graph this system, we need to plot the two equations on the same coordinate plane. The first equation, y1=log⁡0.5xy_1 = \frac{\log 0.5}{x}, is a logarithmic function with a base of 0.5. This function will have a negative slope and will approach the x-axis as x approaches infinity. The second equation, y2=log⁡32+xy_2 = \frac{\log 3}{2 + x}, is a logarithmic function with a base of 3. This function will have a negative slope and will approach the x-axis as x approaches infinity.

The second system of equations is:

y1=log⁡xlog⁡0.5,y2=log⁡(2+x)log⁡3y_1 = \frac{\log x}{\log 0.5}, \quad y_2 = \frac{\log (2 + x)}{\log 3}

To understand this system, let's break it down. The first equation is y1=log⁡xlog⁡0.5y_1 = \frac{\log x}{\log 0.5}. This equation represents a logarithmic function with a base of 0.5 and a variable x. The second equation is y2=log⁡(2+x)log⁡3y_2 = \frac{\log (2 + x)}{\log 3}. This equation represents a logarithmic function with a base of 3 and a variable x.

To graph this system, we need to plot the two equations on the same coordinate plane. The first equation, y1=log⁡xlog⁡0.5y_1 = \frac{\log x}{\log 0.5}, is a logarithmic function with a base of 0.5. This function will have a positive slope and will approach the x-axis as x approaches 0. The second equation, y2=log⁡(2+x)log⁡3y_2 = \frac{\log (2 + x)}{\log 3}, is a logarithmic function with a base of 3. This function will have a positive slope and will approach the x-axis as x approaches 0.

In conclusion, both systems of equations can be graphed to solve the equation log⁡0.5x=log⁡3(2+x)\log_{0.5} x = \log_3 (2 + x). However, the second system of equations, y1=log⁡xlog⁡0.5,y2=log⁡(2+x)log⁡3y_1 = \frac{\log x}{\log 0.5}, \quad y_2 = \frac{\log (2 + x)}{\log 3}, is more suitable for graphing because it has a positive slope and will approach the x-axis as x approaches 0.

Based on the analysis above, the second system of equations, y1=log⁡xlog⁡0.5,y2=log⁡(2+x)log⁡3y_1 = \frac{\log x}{\log 0.5}, \quad y_2 = \frac{\log (2 + x)}{\log 3}, is the most suitable for graphing. This system has a positive slope and will approach the x-axis as x approaches 0, making it easier to graph and find the solution to the equation.

The final answer is B.
Q&A: Which System of Equations Could Be Graphed to Solve the Equation Below?

In our previous article, we explored which system of equations could be graphed to solve the equation log⁡0.5x=log⁡3(2+x)\log_{0.5} x = \log_3 (2 + x). We analyzed two systems of equations and determined that the second system, y1=log⁡xlog⁡0.5,y2=log⁡(2+x)log⁡3y_1 = \frac{\log x}{\log 0.5}, \quad y_2 = \frac{\log (2 + x)}{\log 3}, is the most suitable for graphing.

Q: What is the main difference between the two systems of equations?

A: The main difference between the two systems of equations is the slope of the functions. The first system of equations has a negative slope, while the second system of equations has a positive slope.

Q: Why is the second system of equations more suitable for graphing?

A: The second system of equations is more suitable for graphing because it has a positive slope and will approach the x-axis as x approaches 0. This makes it easier to graph and find the solution to the equation.

Q: What is the significance of the base of the logarithmic functions?

A: The base of the logarithmic functions is significant because it affects the slope of the functions. In the first system of equations, the base is 0.5, which results in a negative slope. In the second system of equations, the base is 3, which results in a positive slope.

Q: How can I determine which system of equations to use?

A: To determine which system of equations to use, you need to analyze the equation and determine the base of the logarithmic functions. If the base is 0.5, use the first system of equations. If the base is 3, use the second system of equations.

Q: What are some common mistakes to avoid when graphing systems of equations?

A: Some common mistakes to avoid when graphing systems of equations include:

  • Not using the correct base of the logarithmic functions
  • Not plotting the functions on the same coordinate plane
  • Not identifying the x-intercept of the functions

Q: How can I find the solution to the equation using the graphed systems of equations?

A: To find the solution to the equation using the graphed systems of equations, you need to identify the x-intercept of the functions. The x-intercept is the point where the two functions intersect, and it represents the solution to the equation.

In conclusion, the second system of equations, y1=log⁡xlog⁡0.5,y2=log⁡(2+x)log⁡3y_1 = \frac{\log x}{\log 0.5}, \quad y_2 = \frac{\log (2 + x)}{\log 3}, is the most suitable for graphing to solve the equation log⁡0.5x=log⁡3(2+x)\log_{0.5} x = \log_3 (2 + x). By understanding the significance of the base of the logarithmic functions and avoiding common mistakes, you can successfully graph the systems of equations and find the solution to the equation.

The final answer is B.