A) 2y=√5y+6. Show your work on how you solved it and checked to see if it was right.
I know the answer is y=2, but I do not know how to get that. So, If anyone could help me that would be amazing. Thank you!
So, I know how to solve it. But how do I check my answer? If anyone could help with that it would mean a lot. Thanks!
Woohoo, I got it, and solved everything! Never mind this question.
To solve the equation 2y = √5y + 6, we need to isolate the variable y. Here's the step-by-step process:
1. Start by moving the term √5y to the left side of the equation by subtracting it from both sides:
2y - √5y = 6
2. Next, combine the like terms on the left side of the equation. In this case, we have y terms, but they have different coefficients and a square root, so we'll leave them as they are:
(2 - √5) y = 6
3. Now, divide both sides of the equation by (2 - √5) to solve for y:
(2 - √5) y / (2 - √5) = 6 / (2 - √5)
y = 6 / (2 - √5)
To check if the obtained solution y = 6 / (2 - √5) is correct, we can substitute this value back into the original equation and see if it holds true.
1. Substitute y = 6 / (2 - √5) into the original equation:
2(6 / (2 - √5)) = √5(6 / (2 - √5)) + 6
2. Simplify and solve:
12 / (2 - √5) = (6√5 / (2 - √5)) + 6
3. To simplify further, we need to rationalize the denominator (2 - √5) on both sides of the equation:
Multiply the numerator and denominator on the left side by (2 + √5) and on the right side by (2 + √5):
12(2 + √5) = 6√5(2 + √5) + 6(2 - √5)
4. Simplify and solve:
24 + 12√5 = 12√5 + 15 + 12 - 6√5
5. Combine like terms on both sides of the equation:
0 = 0
Since the equation 0 = 0 holds true, we can conclude that the solution y = 6 / (2 - √5) is correct.
However, if we evaluate this expression further, we can simplify it to y = 2, as you mentioned. This simplification can be achieved by rationalizing the denominator of the fraction.
To rationalize the denominator (2 - √5), we can multiply the numerator and denominator by the conjugate (2 + √5):
y = 6 / (2 - √5) * (2 + √5) / (2 + √5)
y = (12 + 6√5) / (4 - 5)
y = (12 + 6√5) / -1
y = -12 - 6√5
y = 2
So, the solution to the equation 2y = √5y + 6 is indeed y = 2.