To determine how many solutions the equation has, let's simplify and solve it step by step.
Given the equation:
[tex]\[ 2t - 14t + 2 = -3t + 20 \][/tex]
Step 1: Simplify the equation
Combine like terms on the left-hand side:
[tex]\[ 2t - 14t + 2 = (-12t + 2) \][/tex]
So the equation becomes:
[tex]\[ -12t + 2 = -3t + 20 \][/tex]
Step 2: Move all terms involving [tex]\( t \)[/tex] to one side and constants to the other side
Add [tex]\( 12t \)[/tex] to both sides to move all [tex]\( t \)[/tex]-terms to one side:
[tex]\[ -12t + 12t + 2 = -3t + 12t + 20 \][/tex]
This simplifies to:
[tex]\[ 2 = 9t + 20 \][/tex]
Subtract 20 from both sides to move the constant terms to one side:
[tex]\[ 2 - 20 = 9t + 20 - 20 \][/tex]
This simplifies to:
[tex]\[ -18 = 9t \][/tex]
Step 3: Solve for [tex]\( t \)[/tex]
Divide both sides by 9:
[tex]\[ \frac{-18}{9} = t \][/tex]
[tex]\[ t = -2 \][/tex]
Conclusion
Since we derived a specific value for [tex]\( t \)[/tex], the equation has one solution.
So, the number of solutions is:
[tex]\[
\boxed{1}
\][/tex]
