Translate the two statements from English to math.

For example, "sum" means adding two numbers, "difference" means subtracting two numbers, and "is" means equals. This is how it looks:

#stackrel(x+y) overbrace"The sum of two numbers"" " stackrel= overbrace"is"" " stackrel(51;) overbrace"51."" " stackrel(x-y) overbrace"Their difference"" " stackrel= overbrace"is"" " stackrel13 overbrace"13""."#

Now, make a system of equations:

#color(white)={(x+y=51, qquadqquad(1)), (x-y=13, qquadqquad(2)):}#

Use equation #(2)# to solve for (a temporary) #x# value:

#color(white)=>x-y=13#

#color(white)=>xcolor(red)cancelcolor(black)(-y+y)=13+y#

#color(white)=>x=13+y#

Next, plug this value of #x# into equation #(1)#:

#color(white)=>x+y=51#

#=>(13+y)+y=51#

#color(white)=>13+2y=51#

#color(white)=>2y=51-13#

#color(white)=>2y=38#

#color(white)=>y=38/2#

#color(white)=>y=19#

Lastly, plug this value of #y# into either equation (I'll do #(1)#) and solve for #x#:

#color(white)=>x+y=51#

#=>x+19=51#

#color(white)=>x=51-19#

#color(white)=>x=32#

The two numbers in the word problem are #19# and #32#. Since the problem is asking for the smaller of the two numbers, the answer is #19#.