Question

For what value(s) of k will the following system of linear equations have no solution? infinitely many solutions? x-5y=4-7x+35y=k

Answer 1

x-5y = 4

-7x+35y = k​

Isolating y in both equations,

`\begin{gathered} x=4+5y \\ x-4=5y \\ (1)/(5)x-(4)/(5)=y \end{gathered}`
`\begin{gathered} 35y=k+7x \\ y=(k)/(35)+(7)/(35)x \\ y=(k)/(35)+(1)/(5)x \end{gathered}`

If both equations are equal, then there are infinitely many solutions. This criterion is satisfied if

`\begin{gathered} -(4)/(5)=(k)/(35) \\ -(4)/(5)\cdot35=k \\ -28=k \end{gathered}`

Since both equations have the same slope, if they don't have the same y-intercept then there will be no solution for the system of equations, that is,

`\begin{gathered} -(4)/(5)\\e(k)/(35) \\ -(4)/(5)\cdot35\\e k \\ -28\\e k \end{gathered}`

The given system has no solution for all real numbers k except k= -28

The given system has infinitely many solutions for k= -28