What is the solution to the equation $\frac{1}{x}=\frac{x+3}{2 x^2}$?
A. $x=-3$
B. $x=-3$ and $x=0$
C. $x=0$ and $x=3$
D. $x=3$
Answer (2)
Multiply both sides of the equation by 2 x 2 to get rid of the fractions: 2 x = x + 3 .
Subtract x from both sides to isolate x : x = 3 .
Verify the solution by plugging x = 3 back into the original equation.
The solution to the equation is 3 .
Explanation
Understanding the Problem We are given the equation x 1 = 2 x 2 x + 3 and asked to find its solution. First, we note that x cannot be 0 because division by zero is undefined.
Eliminating Fractions To solve the equation, we can multiply both sides by 2 x 2 to eliminate the fractions. This gives us: 2 x 2 ⋅ x 1 = 2 x 2 ⋅ 2 x 2 x + 3 Simplifying, we get: 2 x = x + 3
Solving for x Now, we solve for x by subtracting x from both sides: 2 x − x = 3 x = 3
Checking the Solution We need to check if our solution is valid by substituting x = 3 back into the original equation: 3 1 = 2 ( 3 2 ) 3 + 3 = 18 6 = 3 1 The solution is valid.
Final Answer Therefore, the solution to the equation is x = 3 .
Examples
Consider a scenario where you are comparing the efficiency of two machines. Machine A completes a task in 'x' hours, while Machine B completes the same task but also handles 3 additional units of work, taking 2 x 2 hours. The equation x 1 = 2 x 2 x + 3 helps determine the value of 'x' for which both machines have equivalent rates of work. Solving this equation allows you to find the time 'x' at which the efficiency of both machines is balanced, aiding in resource allocation and process optimization.
The solution to the equation x 1 = 2 x 2 x + 3 is x = 3 . This was verified by substituting x = 3 back into the original equation, confirming it holds true. Therefore, the answer is option D: x = 3 .
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