Answer:
[tex]\rm 3\; Ag^{1+} + 1\; Al \to 1\; Al^{3+} + 3\; Ag[/tex].
Explanation:
Electrons are conserved in a chemical equation.
The superscript of [tex]\rm Ag^{1+}[/tex] indicates that each of these ions carries a charge of [tex]+1[/tex]. That corresponds to the shortage of one electron for each [tex]\rm Ag^{+}[/tex] ion.
Similarly, the superscript [tex]+3[/tex] on each [tex]\rm Al^{3+}[/tex] ion indicates a shortage of three electrons per such ion.
Assume that the coefficient of [tex]\rm Ag^{+}[/tex] (among the reactants) is [tex]x[/tex], and that the coefficient of [tex]\rm Al^{3+}[/tex] (among the reactants) is [tex]y[/tex].
[tex]\rm \mathnormal{x}\; Ag^{1+} + ?\; Al \to \mathnormal{y}\; Al^{3+} + ?\; Ag[/tex].
There would thus be [tex]x[/tex] silver ([tex]\rm Ag[/tex]) atoms and [tex]y[/tex] aluminum ([tex]\rm Al[/tex]) atoms on either side of the equation. Hence, the coefficient for [tex]\rm Al\![/tex] and [tex]\rm Ag\![/tex] would be [tex]y\![/tex] and [tex]x\![/tex], respectively.
[tex]\rm \mathnormal{x}\; Ag^{1+} + \mathnormal{y}\; Al \to \mathnormal{y}\; Al^{3+} + \mathnormal{x}\; Ag[/tex].
The [tex]x[/tex] [tex]\rm Ag^{1+}[/tex] ions on the left-hand side of the equation would correspond to the shortage of [tex]x[/tex] electrons. On the other hand, the [tex]y[/tex] [tex]Al^{3+}[/tex] ions on the right-hand side of this equation would correspond to the shortage of [tex]3\, y[/tex] electrons.
Just like atoms, electrons are also conserved in a chemical reaction. Therefore, if the left-hand side has a shortage of [tex]x[/tex] electrons, the right-hand side should also be [tex]x\![/tex] electrons short of being neutral. On the other hand, it is already shown that the right-hand side would have a shortage of [tex]3\, y[/tex] electrons. These two expressions should have the same value. Therefore, [tex]x = 3\, y[/tex].
The smallest integer [tex]x[/tex] and [tex]y[/tex] that could satisfy this relation are [tex]x = 3[/tex] and [tex]y = 1[/tex]. The equation becomes:
[tex]\rm 3\; Ag^{1+} + 1\; Al \to 1\; Al^{3+} + 3\; Ag[/tex].
