Question : What is the area (in unit squares) of the triangle enclosed by the graphs of $2 x+5 y=12, x+y=3$ and the x-axis?
Option 1: 2.5
Option 2: 3.5
Option 3: 3
Option 4: 4
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Correct Answer: 3
Solution :
The area of a triangle enclosed by two lines and the x-axis can be found by determining the points where the lines intersect the x-axis and each other.
The line $2x + 5y = 12$ intersects the x-axis when $y = 0$, which gives $x = 6$.
The line $x + y = 3$ intersects the x-axis when $y = 0$, which gives $x = 3$.
The intersection of the lines $2x + 5y = 12$ and $x + y = 3$ can be found by solving these equations simultaneously.
$⇒2(3-y)+5y=12$
$⇒y=2$ and $x=1$
So, the three points of the triangle are $(1,2)$, $(6,0)$, and $(3,0)$.
The base of the triangle is the distance between the points $(6,0)$ and $(3,0)$, which is $3$ units.
The height of the triangle is the y-coordinate of the point $(0,2)$, which is $2$ units.
The area of the triangle = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 2 = 3$ unit squares.
Hence, the correct answer is 3.
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