The Correct Answer and Explanation is:
Of course. Based on the image provided, here is the correct answer and a detailed explanation.
The correct answer is x = 1/3.
This problem requires solving a rational equation, which is an equation containing at least one fraction with a variable in its denominator. The equation given is:
1/x + 2/(3x) = 5
The most effective method for solving this type of equation is to first eliminate the fractions. This is accomplished by finding the least common denominator (LCD) of all the terms and then multiplying the entire equation by it.
First, identify the denominators in the equation: x and 3x. The least common denominator is the smallest expression that both x and 3x can divide into evenly. In this case, the LCD is 3x.
Next, multiply every term in the equation by the LCD, which is 3x:
3x * (1/x) + 3x * (2/(3x)) = 3x * 5
Now, simplify each term. In the first term, the x in the numerator and denominator cancels out, leaving 3 * 1, which is 3. In the second term, the 3x in the numerator and denominator cancels out, leaving just 2. On the right side of the equation, 3x multiplied by 5 is 15x.
This simplifies the equation to a much more manageable linear form:
3 + 2 = 15x
Combine the constants on the left side:
5 = 15x
To solve for x, isolate the variable by dividing both sides of the equation by 15:
x = 5 / 15
Finally, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5.
x = 1/3
The solution is valid because it does not make any of the original denominators equal to zero.
