1. The first step here is to acknowledge the fact that opposite sides of a rectangle are equal in length, ie. the top and bottom sides are the same length, and the left and right sides are the same length. Thus, we can equate the expressions for the opposite lengths to get the following two equations:
(1) Top & bottom: 3x - y = 2x + y
(2) Left & right: 3y + 4 = 2x - 3
2. Now we need to solve the simultaneous equations for x and y. Since you have requested the substitution and elimination methods, I will show these as methods a) and b), respectively. However, first I think it will be worth combining like terms together in each of the equations. Thus:
(1) 3x - y = 2x + y
x - y = y (Subtract 2x from both sides)
x = 2y (Add y to both sides)
(2) 3y + 4 = 2x - 3
3y + 7 = 2x (Add 3 to both sides)
Now that we have combined the like terms in both equations, we can start solving for x and y.
a) Substitution method:
The substitution method relies on our rearranging one of the equations so that it has either x or y as the subject. Since we have combined the like terms together, we have already rearranged equation (1) so that x is the subject, thus we can substitute equation (1) into equation (2):
3y + 7 = 2x
if x = 2y:
3y + 7 = 2(2y) (Substitute x with 2y)
3y + 7 = 4y (2*2y = 4y)
7 = y (Subtract 3y from both sides)
Thus, y = 7. Now, we can substitute this back into equation (1) to find the value of x (note that we could also substitute this into equation 2, I have simply chosen equation 1 since there is less working involved):
x = 2y
if y = 7:
x = 2*7 = 14
Therefor, we have x = 14 and y = 7.
b) The elimination method relies on our 'eliminating' either x or y from the equations by either subtracting them or adding them together. Let us remind ourselves of our equations:
(1) x = 2y
(2) 3y + 7 = 2x
We can see here that we will need to multiply both sides of one, or both, of the equations by a particular value in order to eliminate one of the terms (since in one equation there is x and the other 2x, and in one there is 2y and the other 3y). Choosing to eliminate x will only require us to multiply equation (1) by 2, however if we were to choose to eliminate y, we would have to multiply equation (1) by 3 and equation (2) by 2 (so that each equation has 6y in it - this is their common multiple).
Given that the first option requires less working, I will choose to eliminate x. Thus, if we multiply both sides of equation (1) by 2 we get:
x = 2y
2x = 4y (Multiply both sides by 2)
Thus our two equations now become:
(1) 2x = 4y
(2) 2x = 3y + 7 (here I have simply rearranged equation 2 so that it is simpler to see how the two sides will be subtracted)
Now, we can subtract the left and right sides of each equation from the other (note that we would add them if we had one positive and one negative value so that -2x + 2x = 0).
Thus, subtracting the left and right sides of each equation, we get:
2x - (2x) = 4y - (3y + 7)
0 = 4y - 3y - 7 ( -(3y + 7) = -3y - 7 ; note that in the previous line it is very important to include the brackets around 3y - 7 )
0 = y - 7 (4y - 3y = y)
7 = y (Add 7 to both sides)
Now that we know that y = 7, we can substitute this back into equation (1) to find the value of x:
2x = 4(7)
2x = 28 (4*7 = 28)
x = 14 (28/2 = 14)
Thus, x = 14 and y = 7. This is the same answer that we found when using the substitution method - this is always a good way to check that you are on track.
3. Going back to the question, we can see that we are required to find the values of x and y, as well as the area of the rectangle. Since we have already found that x = 14 and y = 7, we can now calculate the area of the rectangle.
The area of a rectangle is given by the following formula:
A = lw, where l is the length of the rectangle and w is the width.
So far, we do not have the actual length and width of the rectangle, thus this is the next thing we must find. To do so, we must choose one of the expressions for the length (ie. top or bottom) and one for the width (ie. left or right), and substitute x = 14 and y = 7 into them.
I will choose the expression for the bottom length (2x + y) and the expression for the right length (2x - 3). Note that it wouldn't matter which of the expressions for the opposite sides you choose as they are of equal length.
(i) Calculating length:
l = 2x + y
if x = 14 and y = 7:
l = 2(14) + 7
l = 28 + 7
l = 35
(ii) Calculating width:
w = 2x - 3
if x = 14:
w = 2(14) - 3
w = 28 - 3
w = 25
Thus, the length of the rectangle is 35 cm and the width is 25 cm.
Now we are at the final step! Remember that the area of a rectangle may be described by the formula A = lw. We now know that l = 35 and w = 25, thus all that is left is to substitute these two values into the given formula. Thus:
A = lw
if l = 35 and w = 25:
A = 35*25
A = 875 cm^2
4. Thus, x = 14 and y = 7, and the area of the rectangle is 875 cm^2.
