Solve the following equations.Check your result in each case
`t-(2t+5)-5(1-2t)=2(3+4t)-3(t-4)`

Answer

To solve the equation \( t - (2t + 5) - 5(1 - 2t) = 2(3 + 4t) - 3(t - 4) \), we will follow these steps:

### Step 1: Expand the equation
Start by expanding both sides of the equation.

Left-hand side:
\[ t - (2t + 5) - 5(1 - 2t) \]
Distributing the negative sign and the multiplication:
\[ = t - 2t - 5 - 5 + 10t \]
Combine like terms:
\[ = (t - 2t + 10t) + (-5 - 5) = 9t - 10 \]

Right-hand side:
\[ 2(3 + 4t) - 3(t - 4) \]
Distributing:
\[ = 6 + 8t - 3t + 12 \]
Combine like terms:
\[ = (8t - 3t) + (6 + 12) = 5t + 18 \]

Now the equation is:
\[ 9t - 10 = 5t + 18 \]

### Step 2: Move all terms involving \( t \) to one side
Subtract \( 5t \) from both sides:
\[ 9t - 5t - 10 = 18 \]
This simplifies to:
\[ 4t - 10 = 18 \]

### Step 3: Move constant terms to the other side
Add \( 10 \) to both sides:
\[ 4t = 18 + 10 \]
This simplifies to:
\[ 4t = 28 \]

### Step 4: Solve for \( t \)
Divide both sides by \( 4 \):
\[ t = \frac{28}{4} \]
This simplifies to:
\[ t = 7 \]

### Step 5: Check the solution
Substitute \( t = 7 \) back into the original equation to verify:
Left-hand side:
\[ 7 - (2(7) + 5) - 5(1 - 2(7)) = 7 - (14 + 5) - 5(1 - 14) = 7 - 19 - 5(-13) \]
Calculating:
\[ = 7 - 19 + 65 = 53 \]

Right-hand side:
\[ 2(3 + 4(7)) - 3(7 - 4) = 2(3 + 28) - 3(3) = 2(31) - 9 = 62 - 9 = 53 \]

Both sides are equal, so the solution is verified.

### Final Answer:
\[ t = 7 \]