Question

The solution of a pair of linear equations $2x + y = 5$ and $3x + 2y = 8$ will be:

• A. Unique
• B. Two
• C. Infinitely many
• D. None of these

Hint:

For linear equations in two variables, check the ratios $\tfrac{a_1}{a_2}$ and $\tfrac{b_1}{b_2}$. If unequal, the system has a unique solution.

Answer 1

The Correct Option is A

Step 1: Write equations in standard form
\[ 2x + y = 5 $\Rightarrow$ 2x + y - 5 = 0 \] \[ 3x + 2y = 8 $\Rightarrow$ 3x + 2y - 8 = 0 \] So, coefficients are: For first equation: $a_1=2$, $b_1=1$, $c_1=-5$
For second equation: $a_2=3$, $b_2=2$, $c_2=-8$

Step 2: Condition for unique solution
If $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$, then the system has a unique solution.
\[ \frac{a_1}{a_2} = \frac{2}{3}, \frac{b_1}{b_2} = \frac{1}{2} \] Since $\dfrac{2}{3} \neq \dfrac{1}{2}$, the pair of equations has a unique solution.

Step 3: Verification by solving equations
From first equation: $y = 5 - 2x$
Substitute into second equation: \[ 3x + 2(5 - 2x) = 8 \] \[ 3x + 10 - 4x = 8 \] \[ -x + 10 = 8 $\Rightarrow$ x = 2 \] Substitute back: $y = 5 - 2(2) = 1$
So, the unique solution is $(x,y) = (2,1)$.
\[ \boxed{\text{Unique solution at } (2,1)} \]