Question

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method?
$2x+y=5,3x+2y=8$

• A. Unique solutions, $x=2$ and $y=1$
• B. No solutions
• C. Infinitely many solutions
• D. Data insufficient

Answer 1

Answer: (A)

Unique solutions, $x=2$ and $y=1$

Given pair of equations are:
$2x+y=5$ ...(i)
and $3x+2y=8$ ...(ii)
$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}(\frac{a_1}{a_2}=\frac{2}{3},\frac{b_1}{b_2}=\frac{1}{2})$
Here, we have a. unique solution.

By cross multiplication, we have
$\Rightarrow \frac{x}{\left\{\right(1\left)\right(-8)-(2\left)\right(-5\left)\right\}}=\frac{y}{\left\{\right(-5\left)\right(3)-(-8\left)\right(2\left)\right\}}$ $=\frac{1}{\left\{\right(2\left)\right(2)-(3\left)\right(1\left)\right\}}$
$\Rightarrow \frac{x}{(-8+10)}=\frac{y}{(-15+16)}=\frac{1}{(4-3)}$
$\Rightarrow \frac{x}{2}=\frac{y}{1}=\frac{1}{1}$

$\Rightarrow \frac{x}{2}=\frac{1}{1}$ and $\frac{y}{1}=\frac{1}{1}$
$\Rightarrow x=2$ and $y=1$