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------------------------------------------------ Rewrite the following expression:

[tex]\[
\begin{array}{l}
5 \frac{3}{7} = 6 \frac{2}{7} + 1 \frac{3}{7} - 2 \frac{2}{7} \\
1 \frac{9}{12} = \frac{45}{12} - \frac{27}{12} + \frac{3}{12}
\end{array}
\][/tex]

Sure! Let's break down and solve each of the equations step-by-step.

### First Equation
[tex]\[ 5 \frac{3}{7} = 6 \frac{2}{7} + 1 \frac{3}{7} - 2 \frac{2}{7} \][/tex]

#### Step 1: Convert mixed numbers to improper fractions
- [tex]\( 5 \frac{3}{7} = 5 + \frac{3}{7} = \frac{35}{7} + \frac{3}{7} = \frac{38}{7} \)[/tex]
- [tex]\( 6 \frac{2}{7} = 6 + \frac{2}{7} = \frac{42}{7} + \frac{2}{7} = \frac{44}{7} \)[/tex]
- [tex]\( 1 \frac{3}{7} = 1 + \frac{3}{7} = \frac{7}{7} + \frac{3}{7} = \frac{10}{7} \)[/tex]
- [tex]\( 2 \frac{2}{7} = 2 + \frac{2}{7} = \frac{14}{7} + \frac{2}{7} = \frac{16}{7} \)[/tex]

#### Step 2: Simplify the right-hand side
[tex]\[ 6 \frac{2}{7} + 1 \frac{3}{7} - 2 \frac{2}{7} = \frac{44}{7} + \frac{10}{7} - \frac{16}{7} \][/tex]

Combine the fractions:
[tex]\[ \frac{44}{7} + \frac{10}{7} = \frac{54}{7} \][/tex]
[tex]\[ \frac{54}{7} - \frac{16}{7} = \frac{38}{7} \][/tex]

#### Step 3: Verify if left-hand side equals right-hand side
[tex]\[ 5 \frac{3}{7} = \frac{38}{7} \][/tex]
And we saw that the right-hand side simplifies to [tex]\( \frac{38}{7} \)[/tex] as well. Therefore,
[tex]\[ \frac{38}{7} = \frac{38}{7} \][/tex]
This equation holds true.

### Second Equation
[tex]\[ 1 \frac{9}{12} = \frac{45}{12} - \frac{27}{12} + \frac{3}{12} \][/tex]

#### Step 1: Convert mixed number to improper fraction
- [tex]\( 1 \frac{9}{12} = 1 + \frac{9}{12} = \frac{12}{12} + \frac{9}{12} = \frac{21}{12} \)[/tex]

#### Step 2: Simplify the right-hand side
[tex]\[ \frac{45}{12} - \frac{27}{12} + \frac{3}{12} \][/tex]

Combine the fractions:
[tex]\[ \frac{45}{12} - \frac{27}{12} = \frac{18}{12} \][/tex]
[tex]\[ \frac{18}{12} + \frac{3}{12} = \frac{21}{12} \][/tex]

#### Step 3: Verify if left-hand side equals right-hand side
[tex]\[ 1 \frac{9}{12} = \frac{21}{12} \][/tex]
And we saw that the right-hand side simplifies to [tex]\( \frac{21}{12} \)[/tex]. Therefore,
[tex]\[ \frac{21}{12} = \frac{21}{12} \][/tex]
This equation also holds true.

### Conclusion
Both equations are true:
[tex]\[ 5 \frac{3}{7} = 6 \frac{2}{7} + 1 \frac{3}{7} - 2 \frac{2}{7} \][/tex]
[tex]\[ 1 \frac{9}{12} = \frac{45}{12} - \frac{27}{12} + \frac{3}{12} \][/tex]