solve for u, u-5 divide by + 5u+5divide by 6=1

The equation provided is:

\[
\frac{u - 5}{u} + \frac{5u + 5}{6} = 1
\]

Let's solve for \( u \).

### Step 1: Combine the fractions.
First, multiply the entire equation by the common denominator to eliminate the fractions.

The least common denominator between \( u \) and \( 6 \) is \( 6u \). So, multiply the entire equation by \( 6u \):

\[
6u \times \left(\frac{u - 5}{u}\right) + 6u \times \left(\frac{5u + 5}{6}\right) = 6u \times 1
\]

Simplifying, we get:

\[
6(u - 5) + u(5u + 5) = 6u
\]

### Step 2: Expand and simplify the equation.
Expand both terms on the left-hand side:

\[
6u - 30 + 5u^2 + 5u = 6u
\]

Now, combine like terms:

\[
5u^2 + 11u - 30 = 6u
\]

### Step 3: Move all terms to one side.
Subtract \( 6u \) from both sides:

\[
5u^2 + 5u - 30 = 0
\]

### Step 4: Simplify the equation.
Divide the entire equation by 5 to simplify:

\[
u^2 + u - 6 = 0
\]

### Step 5: Factor the quadratic equation.
Now, factor the quadratic equation:

\[
(u + 3)(u - 2) = 0
\]

### Step 6: Solve for \( u \).
Set each factor equal to zero:

\[
u + 3 = 0 \quad \text{or} \quad u - 2 = 0
\]

Solving these, we get:

\[
u = -3 \quad \text{or} \quad u = 2
\]

### Final Answer:
The solutions for \( u \) are \( u = -3 \) and \( u = 2 \).

Would you like a detailed explanation of any of these steps, or do you have any other questions?

Here are some related questions you might find interesting:

1. How do you verify the solutions to a quadratic equation?
2. Can all quadratic equations be factored, or do some require the quadratic formula?
3. What is the significance of the discriminant in solving quadratic equations?
4. How do you solve quadratic equations by completing the square?
5. How do you interpret the solutions of a quadratic equation graphically?

**Tip:** Always check your solutions by plugging them back into the original equation to ensure they satisfy the equation.

Learn how to solve the equation (u - 5)/u + (5u + 5)/6 = 1 step-by-step. This problem involves combining fractions, solving algebraic equations, and applying the quadratic equation formula. Suitable for high school students and beyond.

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