draw the graphs of 2x+y=6 and 2x-y+2=0. shade the region bounded by these lines and x axis. find the area of shaded region
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SOLUTIONS:
We have,
2x + y = 6 _________(i)
2x - y + 2 = 0 _________(ii)
Graph of the equation 2x + y = 6:
We have,
2x + y = 6
y = 6 - 2x
When x = 0, we have y = 6
When x = 3, we have y = 0
Therefore, solutions :- (0, 6), (3, 0)
Plotting these points on graph ( refer to attachment 1 )
We have,
2x - y + 2 = 0
y = 2x + 2
When x = 0, we have y = 2
When x = -1, we have y = 0
Therefore, solutions :- (0, 2) , (-1, 0)
Plotting these points on graph ( refer to attachment 1 )
It is evident from the graph that the two lines intersect at point P(1, 4). The area enclosed by the lines and x-axis is shown in the attachment 1.
Thus, x = 1, y = 4 is the solution of the given system of equations. Draw PM perpendicular from P on x-axis.
Clearly, we have
PM = y-coordinate of point P(1, 4)
PM = 4
DB = 4
∴ Area of the shaded region = Area of ∆PBD
⇒ 1/2( DB × PM )
⇒ 1/2 × 4 × 4
⇒ 8 sq. units.
