Simplify and solve the equation: (4t + 7)(10 - 2t) = 13t - 8t^2 + 40. Find the value of t.

Let's solve this linear equation step-by-step!

Пошаговое решение:

1. Step 1: Expand the left side of the equation.
   We use the distributive property (FOIL method) for (4t + 7)(10 - 2t):
   \( (4t  10) + (4t  -2t) + (7  10) + (7  -2t) \)
   \( 40t - 8t^2 + 70 - 14t \)
2. Step 2: Combine like terms on the left side.
   Combine the 't' terms: \( 40t - 14t = 26t \)
   So the left side becomes: \( -8t^2 + 26t + 70 \)
3. Step 3: Set the expanded left side equal to the right side.
   The equation is now: \( -8t^2 + 26t + 70 = 13t - 8t^2 + 40 \)
4. Step 4: Simplify the equation by moving all terms to one side.
   Notice that we have \( -8t^2 \) on both sides. We can add \( 8t^2 \) to both sides to cancel them out.
   \( 26t + 70 = 13t + 40 \)
5. Step 5: Isolate the 't' terms.
   Subtract \( 13t \) from both sides:
   \( 26t - 13t + 70 = 40 \)
   \( 13t + 70 = 40 \)
6. Step 6: Isolate the constant terms.
   Subtract \( 70 \) from both sides:
   \( 13t = 40 - 70 \)
   \( 13t = -30 \)
7. Step 7: Solve for 't'.
   Divide both sides by \( 13 \):
   \( t = rac{-30}{13} \)

Ответ: t = -30/13