A rectangle has a length of 7m−4n and a width of 5m+6n. Find the equations for the perimeter and the area of the rectangle

To find the equations for the perimeter and area of a rectangle with a given length and width, you can use the following formulas:

1. **Perimeter of a Rectangle:**
   \[
   \text{Perimeter} = 2 \times (\text{Length} + \text{Width})
   \]

2. **Area of a Rectangle:**
   \[
   \text{Area} = \text{Length} \times \text{Width}
   \]

Given:
- Length \( L = 7m - 4n \)
- Width \( W = 5m + 6n \)

### 1. Perimeter Equation

Substituting the values of length and width into the perimeter formula:

\[
\text{Perimeter} = 2 \times \left[(7m - 4n) + (5m + 6n)\right]
\]

First, combine the like terms inside the brackets:

\[
(7m - 4n) + (5m + 6n) = 7m + 5m - 4n + 6n = 12m + 2n
\]

Now, calculate the perimeter:

\[
\text{Perimeter} = 2 \times (12m + 2n) = 24m + 4n
\]

### 2. Area Equation

Substituting the values of length and width into the area formula:

\[
\text{Area} = (7m - 4n) \times (5m + 6n)
\]

To find the area, expand the expression using the distributive property:

\[
\text{Area} = 7m \times 5m + 7m \times 6n - 4n \times 5m - 4n \times 6n
\]

Calculating each term:

\[
\text{Area} = 35m^2 + 42mn - 20mn - 24n^2
\]

Combine like terms:

\[
\text{Area} = 35m^2 + (42mn - 20mn) - 24n^2
\]

\[
\text{Area} = 35m^2 + 22mn - 24n^2
\]

### Summary

- Perimeter equation: \( \text{Perimeter} = 24m + 4n \)
- Area equation: \( \text{Area} = 35m^2 + 22mn - 24n^2 \)

Would you like more details or have any questions?

### Further Questions:

1. How would the perimeter change if the length and width were doubled?
2. What would be the equations if the length and width had different variables (e.g., \( a \) and \( b \))?
3. How does the area equation change if \( m \) and \( n \) are substituted with specific values?
4. Can you find the diagonal of the rectangle using the given length and width?
5. What if one of the dimensions was a constant number instead of an expression?

### Tip:

When dealing with algebraic expressions, always remember to combine like terms and use the distributive property effectively to simplify your calculations.

Learn how to find the equations for perimeter and area of a rectangle with length 7m - 4n and width 5m + 6n. Includes step-by-step solutions and explanations using algebraic expressions and the distributive property. Suitable for Grades 7-9 students.

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