Solve the following equations.Check your result in each case
`13(y-4)-3(y-9)-5(y+4)=0`

Answer

To solve the equation \( 13(y - 4) - 3(y - 9) - 5(y + 4) = 0 \), we will follow these steps:

### Step 1: Expand the equation
We start by distributing the constants across the terms inside the parentheses.

\[ 13(y - 4) = 13y - 52 \]
\[ -3(y - 9) = -3y + 27 \]
\[ -5(y + 4) = -5y - 20 \]

Now substituting these back into the equation gives us:

\[ 13y - 52 - 3y + 27 - 5y - 20 = 0 \]

### Step 2: Combine like terms
Next, we combine all the \(y\) terms and the constant terms.

For the \(y\) terms:
\[ 13y - 3y - 5y = (13 - 3 - 5)y = 5y \]

For the constant terms:
\[ -52 + 27 - 20 = -52 + 27 - 20 = -45 \]

So, we can rewrite the equation as:

\[ 5y - 45 = 0 \]

### Step 3: Solve for \(y\)
Now, we isolate \(y\) by adding 45 to both sides:

\[ 5y = 45 \]

Next, divide both sides by 5:

\[ y = \frac{45}{5} = 9 \]

### Step 4: Check the result
To verify our solution, we substitute \(y = 9\) back into the original equation:

\[ 13(9 - 4) - 3(9 - 9) - 5(9 + 4) = 0 \]

Calculating each term:
\[ 13(5) - 3(0) - 5(13) = 0 \]
\[ 65 - 0 - 65 = 0 \]
\[ 0 = 0 \]

The left-hand side equals the right-hand side, confirming that our solution is correct.

### Final Answer
The solution to the equation is \(y = 9\).

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