Each year, votes are cast for the rookie of the year in a softball league. The voting results for the top three finishers are shown in the table below. How many points is a third-place vote worth?

Let the number of points for a first place vote be x, number of points for a second place vote be y and the number of points for a third place vote be z.

So, by multiplying the number of points by the number of what the point is worth and adding all together, we shall have the total points;

Hence, we have the following equation systems;

23x + 5y + z = 131

5x + 17y + 4z = 80

x + 5y + 15z = 35

These are the three equations we need to solve simultaneously to get the value of z

From equation 3, x = 35 -5y -15z

Insert this into the first two equations;

23 ( 35 -5y -15z) + 5y + z = 131

5(35-5y-15z) + 17y + 4z = 80

805 - 115y -345z + 5y + z = 131

805-131 = 115y + 345z -5y - z

674 = 110y + 344z ••••••••••(*)

The second equation now becomes the following;

5(35-5y-15z) + 17y + 4z = 80

175 -25y -75z + 17y + 4z = 80

175-80 = 25y + 75z -17y -4z

95 = 8y + 71z

8y + 71z = 95 •••••••••••(**)

So we have to solve x and xx simultaneously;

110y + 344z = 674

8y + 71z = 95

110/8 = 13.75 , so multiply equation xx by 13.75 and equation x by 1

110y + 344z = 674

110y + 976.25z = 1306.25

Subtract equation xx from x

(976.25z - 344z) = (1306.25 -674)

632.25z = 632.25

z = 632.25/632.25

z = 1