To find the area of a rectangle, we use the formula:
[tex]\[
\text{Area} = \text{Length} \times \text{Breadth}
\][/tex]
Given the length of the rectangle is [tex]\(5m - 3n\)[/tex] units and the breadth is [tex]\(4m - n\)[/tex] units, we can set up the equation for the area as follows:
[tex]\[
\text{Area} = (5m - 3n) \times (4m - n)
\][/tex]
Now, let's expand this expression step-by-step:
1. Multiply the first term of the length expression by each term in the breadth expression:
[tex]\[
(5m) \times (4m) + (5m) \times (-n)
\][/tex]
This gives:
[tex]\[
20m^2 - 5mn
\][/tex]
2. Multiply the second term of the length expression by each term in the breadth expression:
[tex]\[
(-3n) \times (4m) + (-3n) \times (-n)
\][/tex]
This gives:
[tex]\[
-12mn + 3n^2
\][/tex]
3. Combine all these results together:
[tex]\[
20m^2 - 5mn - 12mn + 3n^2
\][/tex]
4. Simplify by combining like terms:
[tex]\[
20m^2 - 17mn + 3n^2
\][/tex]
Thus, the area of the rectangle, expressed in terms of [tex]\(m\)[/tex] and [tex]\(n\)[/tex], is:
[tex]\[
20m^2 - 17mn + 3n^2
\][/tex]
Putting it all together, the area of the rectangle whose length and breadth are [tex]\(5m - 3n\)[/tex] units and [tex]\(4m - n\)[/tex] units respectively is:
[tex]\[
(5m - 3n) \times (4m - n) = 20m^2 - 17mn + 3n^2
\][/tex]
