Which system of equations could be graphed to solve the equation below?
[tex]$\log _{0.5} x=\log _3 2+x$[/tex]
A. [tex]$y_1=\frac{\log 0.5}{x}, y_2=\frac{\log 3}{2+x}$[/tex]
B. [tex]$y_1=\frac{\log x}{\log 0.5}, y_2=\frac{\log 2+x}{\log 3}$[/tex]
C. [tex]$y_1=\frac{\log 0.5}{\log 3}, y_2=\frac{\log x}{\log 2+x}$[/tex]
D. [tex]$y_1=\frac{\log x}{\log 0.5}, y_2=\frac{\log 2}{\log 3}+x$[/tex]
Answer (1)
Rewrite the left side of the equation as y 1 = lo g 0.5 x = l o g 0.5 l o g x .
Rewrite the right side of the equation as y 2 = lo g 3 2 + x = l o g 3 l o g 2 + x .
The system of equations is y 1 = l o g 0.5 l o g x and y 2 = l o g 3 l o g 2 + x .
The correct system of equations is y 1 = lo g 0.5 lo g x , y 2 = lo g 3 lo g 2 + x .
Explanation
Understanding the Problem We are given the equation lo g 0.5 x = lo g 3 2 + x and asked to find the system of equations that could be graphed to solve it. The idea is to rewrite the given equation as two separate equations, y 1 = f ( x ) and y 2 = g ( x ) , such that the solution to the original equation is the x-coordinate of the intersection point of the graphs of y 1 and y 2 .
Rewriting the Left Side Let's express the left side of the equation as y 1 = lo g 0.5 x . Using the change of base formula, we can rewrite this as y 1 = l o g 0.5 l o g x .
Rewriting the Right Side Now, let's express the right side of the equation as y 2 = lo g 3 2 + x . Using the change of base formula, we can rewrite lo g 3 2 as l o g 3 l o g 2 . Thus, y 2 = l o g 3 l o g 2 + x .
Finding the Correct System of Equations Therefore, the system of equations is y 1 = l o g 0.5 l o g x and y 2 = l o g 3 l o g 2 + x . Comparing this to the given options, we see that the correct answer is y 1 = l o g 0.5 l o g x , y 2 = l o g 3 l o g 2 + x .
Examples
Consider the equation lo g 0.5 x = lo g 3 2 + x . We can solve this equation graphically by plotting two functions y 1 = lo g 0.5 x and y 2 = lo g 3 2 + x and finding their intersection point. This method is useful in various fields, such as engineering and physics, where graphical solutions can provide insights into complex equations and systems. For example, in circuit analysis, the intersection of two curves representing voltage and current can determine the operating point of a circuit.
