Question

The pair of equations 2x + 3y = 5 and 4x + 6y = 15 has ____
(a) a unique solution
(b) exactly two solutions
(c) infinitely many solutions
(d) no solution

Answer 1

The given system of equations can be written as:
$2x+3y-5=0$ and $4x+6y-15=0$
The given equations are of the following form:
$a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$
Here, $a_1=2,b_1=3,c_1=-5,a_2=4,b_2=6$ and $c_2=-15$
$$\begin{array}{cc}& \therefore \frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2},\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2}\text{and}\frac{c_1}{c_2}=\frac{-5}{-15}=\frac{1}{3}\\ & \therefore \frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}\end{array}$$
Hence, the given system has no solution.