Answer:
h=-32 and k=-24
Step-by-step explanation:
[tex]4x_1 + 3x_2 = -1:a_1x_1 + b_1x_2=c_1\\hx_1 +kx_2=8:a_2x_1+b_2x_2=c_2[/tex]
For infinitely many solutions, lines must be same;[tex]\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\\\\=\frac{4}{h}=\frac{3}{k}=\frac{-1}{8}\\\\h=-32, k=-24[/tex]
So, the system gives infinitely many points at [tex](h,k)=(\frac{32}{a},\frac{24}{a} )[/tex].
Given,
[tex]4x_1+3x_2=a\\hx_1+kx_2=8[/tex]
For infinitely many solutions the condition is that,
[tex]\frac{a_1}{a_2}= \frac{b_1}{a_b}=\frac{c_1}{c_2}[/tex]
Then satisfying the given equation into the above condition we get,
[tex]\frac{4}{h}= \frac{3}{k}=\frac{a}{8}\\\frac{4}{h}=\frac{a}{8}\\h=\frac{32}{a} \\ \frac{3}{k}=\frac{a}{8}\\k=\frac{24}{a}[/tex]
So, the point is [tex](h,k)=(\frac{32}{a},\frac{24}{a} )[/tex]
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