# Thread: Infinite Number of Solutions

1. ## Infinite Number of Solutions

Find the value of $\displaystyle m$ for which the following simultaneous equations have an infinite number of solutions:

$\displaystyle 3x+my=5$ and $\displaystyle (m+2)x+5y=m$

So I know that they both need to be the same equation, but I can't work out a process for solving these.

2. $\displaystyle a_1x+b_1y=c_1$ and $\displaystyle a_2x+b_2y=c_2$ have infinite solutions if $\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

From these equations..

$\displaystyle a_1=3,b_1=m$ and $\displaystyle c_1=5$

$\displaystyle a_2=m+2,b_2=5,c_2=m$

so $\displaystyle \frac3{m+2}=\frac m{5}$

$\displaystyle \Rightarrow 15=m^2+2m$

$\displaystyle \Rightarrow m^2+2m-15=0$

$\displaystyle \Rightarrow m=\{-5,3\}$--------------[1]

Also, $\displaystyle \frac m{5}=\frac5{m}$

$\displaystyle \Rightarrow m=\pm 5$-----------------[2]

From equation [1] and equation [2] we get $\displaystyle m=-5$

3. Awesome.
Too easy

Thanks for that!