A. Solve the following rational equations.Note that your solutions will be graded based on the correctness of your LCD (1 point), correctness of the steps prior to the final answer (1 point), correctness of the solutions/roots (1 point) and correctness of the checking part (1 point). (20 points)

3.  3         4             29
    ----   +  ----       =  -------------
   y - 3     y + 2       y^2 - y - 6

Solution Answer:

Checking:

Let's solve the given rational equation step-by-step:

### Given equation:
\[
\frac{3}{y - 3} + \frac{4}{y + 2} = \frac{29}{y^2 - y - 6}
\]

### Step 1: Factor the denominator on the right-hand side
We can factor the quadratic \( y^2 - y - 6 \).

\[
y^2 - y - 6 = (y - 3)(y + 2)
\]

Thus, the equation becomes:

\[
\frac{3}{y - 3} + \frac{4}{y + 2} = \frac{29}{(y - 3)(y + 2)}
\]

### Step 2: Find the Least Common Denominator (LCD)
The LCD of all the denominators is \( (y - 3)(y + 2) \).

### Step 3: Multiply the entire equation by the LCD
We multiply both sides of the equation by \( (y - 3)(y + 2) \) to eliminate the denominators.

\[
(y - 3)(y + 2) \times \frac{3}{y - 3} + (y - 3)(y + 2) \times \frac{4}{y + 2} = (y - 3)(y + 2) \times \frac{29}{(y - 3)(y + 2)}
\]

Simplifying each term:

\[
3(y + 2) + 4(y - 3) = 29
\]

### Step 4: Expand and simplify
Now, expand both sides:

\[
3y + 6 + 4y - 12 = 29
\]

Simplify the left-hand side:

\[
7y - 6 = 29
\]

### Step 5: Solve for \( y \)
Now, solve for \( y \):

\[
7y = 29 + 6
\]
\[
7y = 35
\]
\[
y = \frac{35}{7} = 5
\]

### Step 6: Check the solution
We need to check whether \( y = 5 \) satisfies the original equation:

Substitute \( y = 5 \) into the original equation:

\[
\frac{3}{5 - 3} + \frac{4}{5 + 2} = \frac{29}{5^2 - 5 - 6}
\]

Simplify:

\[
\frac{3}{2} + \frac{4}{7} = \frac{29}{25 - 5 - 6}
\]
\[
\frac{3}{2} + \frac{4}{7} = \frac{29}{14}
\]

Now, find the common denominator for the left-hand side:

\[
\frac{3}{2} + \frac{4}{7} = \frac{3 \times 7}{2 \times 7} + \frac{4 \times 2}{7 \times 2} = \frac{21}{14} + \frac{8}{14} = \frac{29}{14}
\]

Both sides are equal, so the solution \( y = 5 \) is correct.

### Final Answer:
The solution is \( y = 5 \).

Would you like any further details or have any questions?

---

### 5 Related Questions:
1. How do you solve rational equations with more than two terms?
2. What is the significance of finding the LCD when solving rational equations?
3. How do you handle extraneous solutions in rational equations?
4. How does factoring help simplify rational expressions?
5. Can a rational equation have no solution? If so, how do you detect this?

### Tip:
Always check for extraneous solutions by substituting them back into the original equation, especially in rational equations where restrictions on the variable can arise from denominators.

A. Solve the following rational equations. Note that your solutions will be graded based on the correctness of your LCD (1 point), correctness of the steps prior to the final answer (1 point), correctness of the solutions/roots (1 point) and correctness of the checking part (1 point). (20 points)

3 / (y - 3) + 4 / (y + 2) = 29 / (y^2 - y - 6)

Master solving rational equations with this step-by-step guide, featuring a detailed solution to 3 / (y - 3) + 4 / (y + 2) = 29 / (y^2 - y - 6). Learn about factoring quadratics and using the Least Common Denominator (LCD) method to solve complex rational expressions. Perfect for Grades 9-12.

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