Solve the equations and check your answer in the case:
`2(3y-2)-4(2y-5)=9`

Answer

To solve the equation \(2(3y - 2) - 4(2y - 5) = 9\), we will follow these steps:

### Step 1: Expand the equation
We start by distributing the numbers outside the brackets:

\[ 2(3y - 2) - 4(2y - 5) = 9 \]

Distributing \(2\) in the first term:
\[ 2 \times 3y - 2 \times 2 = 6y - 4 \]

Distributing \(-4\) in the second term:
\[ -4 \times 2y + 4 \times 5 = -8y + 20 \]

Now, we can rewrite the equation:
\[ 6y - 4 - 8y + 20 = 9 \]

### Step 2: Combine like terms
Now, we will combine the like terms on the left side of the equation:

\[ (6y - 8y) + (-4 + 20) = 9 \]

This simplifies to:
\[ -2y + 16 = 9 \]

### Step 3: Isolate the variable
Next, we will isolate \(y\) by moving the constant term to the right side of the equation:

\[ -2y = 9 - 16 \]

This simplifies to:
\[ -2y = -7 \]

### Step 4: Solve for \(y\)
Now, we divide both sides by \(-2\):

\[ y = \frac{-7}{-2} = \frac{7}{2} \]

This can also be expressed as:
\[ y = 3.5 \]

### Step 5: Check the solution
To verify our solution, we will substitute \(y = 3.5\) back into the original equation:

\[ 2(3(3.5) - 2) - 4(2(3.5) - 5) = 9 \]

Calculating inside the brackets:
\[ 2(10.5 - 2) - 4(7 - 5) = 9 \]

This simplifies to:
\[ 2(8.5) - 4(2) = 9 \]

Calculating further:
\[ 17 - 8 = 9 \]

Since both sides of the equation are equal, our solution is verified.

### Final Answer
Thus, the value of \(y\) is:
\[ y = \frac{7}{2} \quad \text{or} \quad y = 3.5 \]

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