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)
Start by recognizing that x cannot be zero due to the presence of x in the denominator.
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 .
Conclude that the solution to the equation is 3 .
Explanation
Understanding the Problem We are given the equation x 1 = 2 x 2 x + 3 . Our goal is to find the value(s) of x that satisfy this equation. First, we need to note that x cannot be equal to zero, because division by zero is undefined. So, x = 0 .
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
Isolating x Now, we want to isolate x . We can subtract x from both sides of the equation: 2 x − x = x + 3 − x This simplifies to: x = 3
Checking the Solution Since our solution is x = 3 , and we know that x cannot be zero, this is a valid solution. Therefore, the solution to the equation is x = 3 .
Examples
Imagine you are designing a seesaw where the balance depends on the position of the fulcrum. The equation x 1 = 2 x 2 x + 3 can be analogous to finding the correct position x (distance from a reference point) to balance the seesaw given certain weight distributions. Solving such equations helps engineers and designers determine precise measurements to ensure stability and equilibrium in various mechanical systems, from simple levers to complex structures.
The solution to the equation x 1 = 2 x 2 x + 3 is x = 3 . This solution is valid as it does not violate the restriction that x = 0 . Thus, the correct answer is option D: x = 3 .
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