Certainly, I can help you solve the equation step by step:
2x + 3/(2x + x) + 5y = 180
First, simplify the denominator inside the fraction:
2x + 3/(3x) + 5y = 180
Now, let's find a common denominator for the terms involving fractions, which is 3x:
(2x * 3x) + (3 * 1) + 5y(3x) = 180 * 3x
This simplifies further:
6x^2 + 3 + 15xy = 540x
Now, let's move all the terms to one side of the equation to set it equal to zero:
6x^2 + 15xy - 540x + 3 = 0
Now, we have a quadratic equation in terms of x. To solve for x, we can either factor it or use the quadratic formula. In this case, factoring might be a bit complicated, so let's use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Where:
a = 6
b = 15y
c = 3 - 540y
Now, substitute these values into the formula:
x = (-15y ± √((15y)² - 4 * 6 * (3 - 540y))) / (2 * 6)
Simplify further:
x = (-15y ± √(225y² - 4 * 6 * (3 - 540y))) / 12
Now, simplify within the square root:
x = (-15y ± √(225y² - 12(3 - 540y))) / 12
x = (-15y ± √(225y² - 36 + 6480y)) / 12
x = (-15y ± √(225y² + 6480y - 36)) / 12
Now, you can simplify further, but it's important to note that this equation might have two solutions for x due to the ± sign.