I have these two equations: $cx+y=5, x+y=2$. For what $c$ would this have no solution, infinite solution, unique solution.
For no solution I got when c=1, and for c=0, we have unique solution. Is this right, how do I find where it has infinite solutions.
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Use that if $\frac{a_1}{a_2}\neq \frac{b_1}{b_2}$, the equations will have a unique solution, where $a_1,a_2,b_1,b_2$ are the coefficients. So for $\frac{c}{1}\neq\frac{1}{1}$ the equations will have a unique solution.
If $\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$, the equations will be inconsistent. So $\frac c1=\frac11\neq\frac52$. So for $c=1$, equations will be parallel.
For $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$, the equations will have infinite solutions. But as we see $c=1=\frac52$ is not true. So for no real value of $c$, this condition is satisfied.
If we subtract the first and the second equation, we note that $$(c-1)x=3$$ This means that, for $c=1$ it hasn't solution ($0=3$) and for $c\neq1$ it has only one solution. Here, there isn't case of infinite solution because with the subtraction of the two equations we have not obtained $0=0$.