To find the value of [tex]\( m \)[/tex], let's start by understanding the perimeters of the hexagon and the pentagon.
### Hexagon
A regular hexagon has 6 equal sides. The length of each side of the hexagon is given by [tex]\( 7m + 3 \)[/tex]. Therefore, the perimeter of the hexagon can be calculated as:
[tex]\[
\text{Perimeter of Hexagon} = 6 \times (7m + 3)
\][/tex]
### Pentagon
A regular pentagon has 5 equal sides. The length of each side of the pentagon is given by [tex]\( 8m + 6 \)[/tex]. Therefore, the perimeter of the pentagon can be calculated as:
[tex]\[
\text{Perimeter of Pentagon} = 5 \times (8m + 6)
\][/tex]
### Setting the Perimeters Equal
Since the perimeter of the hexagon is equal to the perimeter of the pentagon, we set the two expressions for the perimeter equal to each other:
[tex]\[
6 \times (7m + 3) = 5 \times (8m + 6)
\][/tex]
### Simplifying the Equation
First, expand both sides:
[tex]\[
6 \times 7m + 6 \times 3 = 5 \times 8m + 5 \times 6
\][/tex]
[tex]\[
42m + 18 = 40m + 30
\][/tex]
Next, isolate the terms involving [tex]\( m \)[/tex]:
[tex]\[
42m + 18 - 40m = 30
\][/tex]
[tex]\[
2m + 18 = 30
\][/tex]
Subtract 18 from both sides to solve for [tex]\( m \)[/tex]:
[tex]\[
2m = 30 - 18
\][/tex]
[tex]\[
2m = 12
\][/tex]
Finally, divide by 2:
[tex]\[
m = \frac{12}{2}
\][/tex]
[tex]\[
m = 6
\][/tex]
### Conclusion
The value of [tex]\( m \)[/tex] is [tex]\( \boxed{6} \)[/tex].
