is there like a way where I could do the butterfly method multiplication to solve this rational equation $$ \frac{1}{h+5}-2=\frac{3}{h+5} $$

Yes, the butterfly method (also called cross-multiplication) can be applied to rational equations, but in this particular case, we can simplify the process even more since the denominators are already the same. Here’s how you can solve the equation step by step:

The given equation is:
\[
\frac{1}{h+5} - 2 = \frac{3}{h+5}
\]

### Step 1: Move all terms with the same denominator to one side.
Since both fractions have the same denominator $ h + 5 $, you can subtract $\frac{1}{h+5}$ from $\frac{3}{h+5}$. So, first, move $\frac{1}{h+5}$ to the right side:

\[
-2 = \frac{3}{h+5} - \frac{1}{h+5}
\]

### Step 2: Combine the fractions on the right-hand side.
Since the denominators are the same, you can combine the numerators:

\[
-2 = \frac{3 - 1}{h+5}
\]
\[
-2 = \frac{2}{h+5}
\]

### Step 3: Isolate the variable.
Now you can use cross-multiplication (which is a part of the butterfly method) to get rid of the denominator:

\[
-2(h+5) = 2
\]

### Step 4: Simplify the equation.
Distribute the $-2$ on the left-hand side:

\[
-2h - 10 = 2
\]

### Step 5: Solve for $h$.
Now, solve for $h$:

\[
-2h = 2 + 10
\]
\[
-2h = 12
\]
\[
h = \frac{12}{-2}
\]
\[
h = -6
\]

### Final Answer:
The solution to the equation is:
\[
h = -6
\]

Would you like further details or clarifications on any of the steps?

---

### Follow-up Questions:
1. How can we check if our solution $ h = -6 $ is valid by substituting it back into the original equation?
2. What would the process look like if the denominators were different in the rational equation?
3. How does cross-multiplication help in solving rational equations generally?
4. What happens if $h + 5 = 0$ in this equation?
5. Can we use the butterfly method for non-linear rational equations?

### Tip:
Always check your final answer by substituting it back into the original equation to ensure there are no extraneous solutions, especially in rational equations.

Is there like a way where I could do the butterfly method multiplication to solve this rational equation $$ \frac{1}{h+5}-2=\frac{3}{h+5} $$?

Learn how to solve the rational equation \frac{1}{h+5} - 2 = \frac{3}{h+5} step by step using the cross multiplication method. This solution covers simplifying rational expressions and solving for the variable h. Suitable for students in Grades 8-10.

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