To solve the equation \( \frac{7}{9} - \frac{3}{4}b = \frac{193}{144} \), follow these steps:
-
Isolate the term with \( b \):
Start by moving \( \frac{7}{9} \) to the right side of the equation: \[ -\frac{3}{4}b = \frac{193}{144} - \frac{7}{9} \]
-
Get a common denominator:
The denominators are 144 and 9. The least common multiple of 144 and 9 is 144. We can convert \( \frac{7}{9} \) to have a denominator of 144: \[ \frac{7}{9} = \frac{7 \times 16}{9 \times 16} = \frac{112}{144} \]
Now, substitute this back into the equation: \[ -\frac{3}{4}b = \frac{193}{144} - \frac{112}{144} \] \[ -\frac{3}{4}b = \frac{193 - 112}{144} = \frac{81}{144} \]
-
Simplify the fraction:
Reduce \( \frac{81}{144} \): \[ \frac{81 \div 9}{144 \div 9} = \frac{9}{16} \]
Now, the equation is: \[ -\frac{3}{4}b = \frac{9}{16} \]
-
Solve for \( b \):
To solve for \( b \), multiply both sides by \(-\frac{4}{3}\): \[ b = \frac{9}{16} \times -\frac{4}{3} \] \[ b = -\frac{9 \cdot 4}{16 \cdot 3} = -\frac{36}{48} \]
-
Simplify \( -\frac{36}{48} \): \[ \frac{36 \div 12}{48 \div 12} = -\frac{3}{4} \]
Thus, the solution to the equation is: \[ \boxed{-\frac{3}{4}} \]