Show why the below equation is true using the areas of rectangles:
2p+5p=(2+5)p
3(4-1)=3x4-3x1
To prove the equation 2p + 5p = (2 + 5)p using the areas of rectangles, we can visualize the equation as a representation of two different rectangles.
First, let's consider the left side of the equation, 2p + 5p. To represent this visually, we can imagine two rectangular shapes. One rectangle has a length of 2 units and a width of p units, while the other rectangle has a length of 5 units and a width of p units.
The area of a rectangle is calculated by multiplying its length by its width. So, we can calculate the area of each rectangle as follows:
Area of the first rectangle = Length × Width = 2 × p = 2p
Area of the second rectangle = Length × Width = 5 × p = 5p
Now, let's move on to the right side of the equation, (2 + 5)p. Here, we have a single rectangle with a length of (2 + 5) units and a width of p units.
To calculate the area of this rectangle, we multiply the length (the sum of 2 and 5) by the width (p):
Area of the rectangle = (2 + 5) × p = 7 × p = 7p
Hence, for the equation to be true, the areas of the two sets of rectangles must be equal:
Area of the left side = Area of the right side
2p + 5p = 7p
Therefore, the equation 2p + 5p = (2 + 5)p holds true based on the concept of the areas of rectangles.
Moving on to the second equation, 3(4 - 1) = 3 × 4 - 3 × 1, let's similarly use the areas of rectangles to illustrate the relationship between the two sides.
On the left side of the equation, we have 3(4 - 1), which represents a rectangle with a length of 3 units and a width of (4 - 1) units. The area of this rectangle can be calculated as:
Area = Length × Width = 3 × (4 - 1) = 3 × 3 = 9
On the right side of the equation, we have 3 × 4 - 3 × 1, which can be simplified as:
3 × 4 = 12
3 × 1 = 3
12 - 3 = 9
As we can see, the area of the rectangle in both cases is 9, confirming that the equation is true:
Area of the left side = Area of the right side
3(4 - 1) = 3 × 4 - 3 × 1
Therefore, the equation 3(4 - 1) = 3 × 4 - 3 × 1 is also true based on the areas of rectangles.