To solve the problem, we need to find the lengths of two adjacent sides of the parallelogram. The given lengths of the sides are expressed in terms of \( n \).
First, let's consider the two opposite sides of the parallelogram:
[tex]\[ 5n - 6 \quad \text{(side 1)} \][/tex]
[tex]\[ 3n - 2 \quad \text{(side 2)} \][/tex]
For these lengths to represent the opposite sides of a parallelogram, they must be equal. Thus, we set the two expressions equal to each other and solve for \( n \):
[tex]\[ 5n - 6 = 3n - 2 \][/tex]
Subtract \( 3n \) from both sides to isolate \( n \):
[tex]\[ 5n - 3n - 6 = -2 \][/tex]
[tex]\[ 2n - 6 = -2 \][/tex]
Next, add 6 to both sides:
[tex]\[ 2n - 6 + 6 = -2 + 6 \][/tex]
[tex]\[ 2n = 4 \][/tex]
Divide both sides by 2 to solve for \( n \):
[tex]\[ n = 2 \][/tex]
Now that we have the value of \( n \), we can find the lengths of the sides. Substitute \( n = 2 \) into the expressions for the sides:
1. Substitute \( n = 2 \) into \( 5n - 6 \) (which is one of the opposite sides):
[tex]\[ 5(2) - 6 = 10 - 6 = 4 \][/tex]
2. Substitute \( n = 2 \) into \( 2n + 3 \) (the third side):
[tex]\[ 2(2) + 3 = 4 + 3 = 7 \][/tex]
The lengths of two adjacent sides of the parallelogram are:
[tex]\[ 4 \, \text{cm} \text{ and } 7 \, \text{cm} \][/tex]
Therefore, the correct answer is:
[tex]\[ 4 \, \text{cm} \text{ and } 7 \, \text{cm} \][/tex]
So, the option is:
[tex]\[ \boxed{4 \, \text{cm} \text{ and } 7 \, \text{cm}} \][/tex]
