To solve the equation ( x - 7 \frac{4}{5} = -4 \frac{2}{9} ), we first convert the mixed numbers into improper fractions.
Convert ( 7 \frac{4}{5} ):
[
7 \frac{4}{5} = 7 \times 5 + 4 = 35 + 4 = 39 \frac{4}{5} = \frac{39 \times 5 + 4}{5} = \frac{39 + 4}{5} = \frac{199}{5}
]
Convert ( -4 \frac{2}{9} ):
[
-4 \frac{2}{9} = -\left(4 \times 9 + 2\right) = -\left(36 + 2\right) = -38
]
So now we have:
[
x - \frac{39}{5} = -\frac{38}{9}
]
Next, we need to add ( \frac{39}{5} ) to both sides to isolate ( x ):
[
x = -\frac{38}{9} + \frac{39}{5}
]
To perform this addition, we need a common denominator, which is ( 45 ).
Convert ( -\frac{38}{9} ) to the common denominator:
[
-\frac{38}{9} = -\frac{38 \times 5}{45} = -\frac{190}{45}
]
Convert ( \frac{39}{5} ) to the common denominator:
[
\frac{39}{5} = \frac{39 \times 9}{45} = \frac{351}{45}
]
Now we can add the fractions:
[
x = -\frac{190}{45} + \frac{351}{45} = \frac{351 - 190}{45} = \frac{161}{45}
]
Thus, the solution is:
[
x = \frac{161}{45}
]
You can also convert this back to a mixed number, if desired:
[
\frac{161}{45} = 3 \frac{26}{45}
]
So, the final answer is:
[
x = \frac{161}{45} \quad \text{or} \quad x = 3 \frac{26}{45}
]
