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.
