Find the area of the shaded region.
In order to calculate the area of the shaded region, we need to subtract the area of the triangle from the area of the rectangle. The area of any rectangle can be calculated by multiplying the length by the width. The length of the rectangle is seven 𝑥 plus 10𝑦. And the width is 10 multiplied by 𝑥 minus 𝑦. This can be simplified to 10𝑥 minus 10𝑦 by expanding or multiplying out the bracket or parenthesis.
The area of the rectangle is, therefore, equal to seven 𝑥 plus 10𝑦 multiplied by 10𝑥 minus 10𝑦. We can expand these two parentheses using the FOIL method. Multiplying the first terms, seven 𝑥 and 10𝑥, gives us 70𝑥 squared. Multiplying the outside terms, seven 𝑥 and negative 10𝑦, gives us negative 70𝑥𝑦. Multiplying the inside terms, 10𝑦 and 10𝑥, gives us 100𝑥𝑦. And finally, multiplying the last terms, 10𝑦 and negative 10𝑦, gives us negative 100𝑦 squared.
Collecting, or grouping, the like terms negative 70𝑥𝑦 plus 100𝑥𝑦, gives us 30𝑥𝑦. Therefore, the area of the rectangle is equal to 70𝑥 squared plus 30𝑥𝑦 minus 100𝑦 squared.
Our next step is to calculate the area of the triangle. In order to calculate the area of any triangle, we multiply the base by the perpendicular height and divide the answer by two. The base of the triangle is five multiplied by 𝑥 plus 𝑦. Expanding this bracket, or parenthesis, gives us five 𝑥 plus five 𝑦.
The height of the triangle is the same as the height, or width, of the rectangle, 10𝑥 minus 10𝑦. Once we’ve expanded and simplified these two parentheses, we need to divide the answer by two to calculate the area of the triangle.
Once again, we can expand the two parentheses using the FOIL method. Multiplying the first terms gives us 50𝑥 squared. Multiplying the outside terms gives us negative 50𝑥𝑦. Multiplying the inside terms gives us positive 50𝑥𝑦. And multiplying the last terms gives us negative 50𝑦 squared. This is all divisible by two.
Negative 50𝑥𝑦 plus 50𝑥𝑦 is equal to zero. So, these two terms cancel. We’re, therefore, left with 50𝑥 squared minus 50𝑦 squared divided by two. As 50 divided by two, or a half of 50, is equal to 25, the area of the triangle is equal to 25𝑥 squared minus 25𝑦 squared.
We now need to subtract the area of the triangle from the area of the rectangle. 70𝑥 squared plus 30𝑥𝑦 minus 100𝑦 squared minus 25𝑥 squared minus 25𝑦 squared. In order to simplify this, we’ll once again collect the like terms. Firstly, 70𝑥 squared minus 25𝑥 squared. 70 minus 25 is equal to 45. So, we have 45𝑥 squared. There is only one 𝑥𝑦 term, 30𝑥𝑦.
Next, we need to group, or collect, the 𝑦 squared terms. Negative 100𝑦 squared minus negative 25𝑦 squared. The two negatives will turn into an addition sign, or a positive. So, we, therefore, have negative 100𝑦 squared plus 25𝑦 squared. This is equal to negative 75𝑦 squared, as negative 100 plus 25 equals negative 75. This means that the area of the shaded region is 45𝑥 squared plus 30𝑥𝑦 minus 75𝑦 squared.