Answer to Question #232429 in Linear Algebra for Phummy

Question

You are given the system of linear equations

 2x+ky=5,  x+3y=7,

 

where k

k is a constant.

The system above has no solution when k=

Answer 1

$2x+ky=5\Rightarrow 2x+ky-5=0 ...(1) \\ x+3y=7\Rightarrow x+3y-7=0 ...(2) \\Comparing\ (1)\ with \ a_1x+b_1y+c_1=0\ and\ (2)\ with \ a_2x+b_2y+c_2=0 \\a_1=2,b_1=k,c_1=-5,a_2=1,b_2=3,c_2=-7$

For no solution,

$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2} \\\frac{2}{1}=\frac{k}{3}\neq\frac{-5}{-7} \\\Rightarrow k=6, k\neq \frac {15}{7}$