In 2008, a cure for cancer walk sold 21 pink sponsorships and 10 gold sponsorships for a total of $770. in 2009, the walk collected $840 by selling 7 pink sponsorships and 20 gold sponsorships. what is the cost of one pink sponsorship?

1.
Let the cost of one pink sponsorship be P dollars, and the cost of one gold sponsorship be G dollars.

2.
"In 2008, a cure for cancer walk sold 21 pink sponsorships and 10 gold sponsorships for a total of $770"

means that

21*P ($ per pink sponsorship)+10*G ($ per pink sponsorship)= $ 770,

thus, i) 21P+10G=770

3.

"In 2009, the walk collected $840 by selling 7 pink sponsorships and 20 gold sponsorships."

means that 

7*P ($ per pink sponsorship)+20*G ($ per pink sponsorship)= $ 840,

thus, ii) 7P+20G=840

4.
we find P and G by solving the system of linear equations:

i) 21P+10G=770
ii) 7P+20G=840

notice the G's:  20 G = (-2) * 10G

multiply equation i) by -2:   -42P - 20G = -1540 and add it to equation ii)

 i) -42P - 20G = -1540
ii)   7P + 20G = 840

       -42P+7P = -1540+840
              -35P = -700
                   P=-700/(-35)= 20          ($)

The cost of one pink sponsorship is $20

Remark: we used the "method by elimination", to solve the system of equations.