T he area of a rectangle gets reduced by 9 square units if its length is reduced by 5 units and breadth is increased by 3 units . if we increase the length of 3 units and the breadth by 2 units the area increases by 67 square units . find the dimensions of the rectangle .
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Answers
Answer:
Let the length and breadth of the rectangle be x and y units respectively. Then,
Area =xy sq. units.
If length is reduced by 5 units and the breadth is increases by 3 units, then area is reduced by 9 square units.
∴xy−9=(x−5)(y+3)
⇒xy−9=xy+3x−5y−15
⇒3x−5y−6=0 ...(i)
When length is increased by 3 units and breadth by 2 units, the area is increased by 67 sq. units.
∴xy+67=(x+3)(y+2)
⇒xy+67=xy+2x+3y+6
⇒2x+3y−61=0 ...(ii)
Thus, we get the following system of linear equations:
3x−5y−6=0
2x+3y−61=0
By using cross-multiplication, we have
305+18
x
=
−183+12
−y
=
9+10
1
⇒x=
19
323
=17 and y=
19
171
=19
Hence, the length and breadth of the rectangle are 17 units and 19 units respectively.
We know that the area of rectangle is of the form xy where length=x and breadth=y
Now according to question,
(x-5)(y+3) = xy-9 -- (i)
(x+3)(y+2) = xy + 67--- (ii)
On solving the 2 equations,we get
xy+3x-5y-15 = xy - 9
→ 3x-5y = 6 -- (iii)
→ xy+2x+3y+6 = xy + 67
→ 2x + 3y =61 (iv)
→ 6x-10y = 12(v)
→ 6x+9y=183(vi)
On subtracting (vi) from (v),we get
-19y = -171
→ y=9
On substituting y=9 in (vi),we get
6x + 81=183
→ 6x = 102 (6x+9y=183 where y=9 is substituted in this equation)
So, x=17
Therefore,the dimensions of the rectangle are:
Length(x) = 17 units
Breadth(y) = 9 units
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