[SOLVED]311.1.3.19 find h and k in the system

karush

Well-known member
$\tiny{311.1.3.19}$
find h and k such that the system has, no solution (parallel) ,unique solution (intersection) , many solutions (same line)
$\begin{array}{rrr} x_1+hx_2&=2&(1)\\ 4x_2+8x_2&=k&(2) \end{array}$

by observation if h=2 then $k=\mathbb{R}, k\ne 8$ for parallel

if $h\ne 2$ then intersection

if $h=2$ and $k=8$ many solutions

Country Boy

Well-known member
MHB Math Helper
I presume that you solved the equations:
Multiply the first equation by 4 to get $$\displaystyle 4x_1+ 4hx_2= 8$$.
Subtract that from the second equation, $$\displaystyle 4x_1+ 8x_2= k$$ to get $$\displaystyle (8- 4h)x_2= k- 8$$.

Solve for $$\displaystyle x_2$$ by dividing by 8- 4h: $$\displaystyle x_2= \frac{k- 8}{8- 4h}$$. As long as 8- 4h is not 0, which means h is not 2, there is a unique answer. (There are two intersecting lines.)

If h= 2 and k= 8 the equation is $$\displaystyle 0x_2= 0$$ which is true for all $$\displaystyle x_2$$ so there are infinitely many solutions. (There is only one line described by both equations.)

If h= 2 and k is not 8, the equation is $$\displaystyle 0x_2= k- 8$$ which is NOT true for any $$\displaystyle x_2$$. There is no solution. (There are two parallel lines.)

That is what you have. Well done!

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