To find the value of \( y \) that makes the expression \( yb - 6 + 7 + 2b - 3b + 1 \) equivalent to \( 5b + 2 \), we simplify the left-hand side first:
Start with the original expression:
\[ yb - 6 + 7 + 2b - 3b + 1 \]
Now combine like terms:
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Combine the constant terms:
\[ -6 + 7 + 1 = 2 \]
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Combine the \( b \) terms:
\[ yb + 2b - 3b = (y + 2 - 3)b = (y - 1)b \]
Putting it all together, we get:
\[ (y - 1)b + 2 \]
Next, we set this equal to the right-hand side, \( 5b + 2 \):
\[ (y - 1)b + 2 = 5b + 2 \]
To solve for \( y \), we can first subtract 2 from both sides:
\[ (y - 1)b = 5b \]
Next, we can factor out \( b \) from both sides, assuming \( b \neq 0 \):
\[ y - 1 = 5 \]
Now solve for \( y \):
\[ y = 5 + 1 \]
\[ y = 6 \]
Thus, the value of \( y \) that makes the expression equivalent to \( 5b + 2 \) is
\[ \boxed{6} \]
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