I have $2$ equations :
$$-\frac{a}{2}x-\frac{b}{2}+cx+d=x+2$$
$$-2ax-2b+cx+d=2x+1$$
with $a, b, c, x \neq 0$
We have to find all possible solutions of $a, b, c, d$ that make the equations true for all $x$.
I found one solution
$a=-\frac{2}{3}$,
$b=\frac{2}{3}$,
$c=\frac{2}{3}$,
$d=\frac{7}{3}$
But, I wonder if there are any other solutions to this system of equations?
$\endgroup$ 32 Answers
$\begingroup$Write the equations as:
$$ \begin{cases} \begin{align} (a-2c+2)x+b-2d+4 = 0 \\ (2a-c+2)x+2b-d+1 = 0 \end{align} \end{cases} $$
A polynomial is identical $0$ (i.e. for all $x$) iff all its coefficients are $0\,$, which gives the system to solve:
$$ \begin{cases} \begin{align} a - 2c + 2 = 0 \\ 2a - c + 2 = 0 \\[7px] b - 2d + 4 = 0 \\ 2b -d + 1 = 0 \end{align} \end{cases} $$
The system has the unique solution as posted $\;a=-\frac{2}{3}$, $b=\frac{2}{3}$, $c=\frac{2}{3}$, $d=\frac{7}{3}\,$.
$\endgroup$ $\begingroup$Your $2\times 2$ system is consisted of $5$ variables (unknowns) to which you're being asked to find the solutions. Since you have only two equations for $5$ unknowns, this means that there is an infinite number of solutions (infinite number of numbers $a,b,c,d,x$ that satisfy the system given).
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