Respuesta :
Let k be the number in the blank, so that:
[tex]7x+3=7x-k[/tex]Substract 7x from both sides:
[tex]3=-k[/tex]These two equations are equivalent regardless the value of x. We can change the conclusions that we may obtain by choosing different values for k.
Then, the equation:
[tex]7x+3=7x-0[/tex]Is true for no values of x.
If we want the equation to be false regardless of the value of x, then set k so that -k is different from 3. For example, set k=0:
[tex]\begin{gathered} 3=-0 \\ \Rightarrow3=0 \end{gathered}[/tex]Since this is contradictory, then there are no values of x that make the equation true.
If we want the equation to be true for all values of x, then 3=-k must be an identity. Then, let k=-3:
[tex]\begin{gathered} 3=-(-3) \\ \Rightarrow3=3 \end{gathered}[/tex]Then, the equation:
[tex]7x+3=7x-(-3)[/tex]Is true for all values of x.
If we want the equation to be true for only one value of x, we have to bring back x into the equation 3=-k. So, we can take k=x. This way, we would have:
[tex]\begin{gathered} 7x+3=7x-x \\ \Rightarrow3=-x \\ \Rightarrow x=-3 \end{gathered}[/tex]