Exercise 2.10 - Chapter 2 - Algebra - 11th Maths Guide Samacheer Kalvi Solutions

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Ex 2.10
Question 1.
$\mathrm{x} \leq 3 \mathrm{y}, \mathrm{x} \geq \mathrm{y}$
Solution:
Given in equation are $x \leq 3 y, x \geq y$
Suppose $\mathrm{x}=3 \mathrm{y} \Rightarrow \frac{x}{3}==\mathrm{y}$

If $x=y$

Question 2.
$
y \geq 2 x,-2 x+3 y \leq 6
$
Solution:
Suppose $\mathrm{y}=2 \mathrm{x}$

$
\begin{aligned}
& -2 x+3 y=6 \\
& -2 x=6-3 y
\end{aligned}
$
$
\begin{aligned}
& x=\frac{6-3 y}{-2} \\
& x=-\frac{6}{2}-\frac{3 y}{-2} \\
& x=-3+\frac{3 y}{2} \\
& 3+x=\frac{3 y}{2}
\end{aligned}
$

Question 3.
$
3 x+5 y \geq 45, x \geq 0, y \geq 0 \text {. }
$
Solution:
If $3 x+5 y=45$

$x \geq 0$ is nothing but the positive portion of $Y$-axis and $y \geq 0$ is the positive portion of $X$-axis. Shaded region is the required portions.
Question 4.
$
2 x+3 y \leq 35, y \geq 2, x \geq 5
$
Solution:
If $2 x+3 y=35$ then

$y=2$ is a line parallel to $X$-axis at a distance 2 units
$x=5$ is a line parallel to $Y$-axis at a distance of 5 units

The required region is below $2 x+3 y=35$, above $y=2$ and to the right of $x=5$
Question 5.
$
2 x+3 y \leq 6, x+4 y \leq 4, x \geq 0, y \geq 0 \text {. }
$
Solution:
If $2 x+3 y=6$

$
x+4 y=4
$

$x \geq 0, y \geq 0$ represents the area in the 1 quadrant.

The required region is below $2 \mathrm{x}+3 \mathrm{y}=6$ and below $\mathrm{x}+4 \mathrm{y}=4$ bounded by $\mathrm{x}$-axis and $\mathrm{y}$-axis.
Question 6.
$
x-2 y \geq 0,2 x-y \leq-2, x \geq 0, y \geq 0
$
Solution:
If $x-2 y=0$

2 x-y=-2

$x \geq 0, y \geq 0$ represents the portion in the 1 quadrant only.
Question 7.
$2 x+y \geq 8, x+2 y \geq 8, x+y \leq 6$.
Solution:
$2 x+y=8$

x+2 y=8

x+y=6