Given:
Two small pizzas, a liter of soda, and a salad costs $14.
One small pizza, a liter of soda, and three salads costs $15.
Three small pizzas, a liter of soda, and two salads costs $22.
To find:
The cost of each item.
Solution:
Let x be the cost of a small pizza.
y be the cost of a liter of soda
z be the cost of a salad.
According to the question, two small pizzas, a liter of soda, and a salad costs $14.
[tex]2x+y+z=14[/tex] ...(i)
One small pizza, a liter of soda, and three salads costs $15.
[tex]x+y+3z=15[/tex] ...(ii)
Three small pizzas, a liter of soda, and two salads costs $22.
[tex]3x+y+2z=22[/tex] ...(iii)
From (i), we get
[tex]z=14-2x-y[/tex] ...(iv)
Putting [tex]z=14-2x-y[/tex] in (ii), we get
[tex]x+y+3(14-2x-y)=15[/tex]
[tex]x+y+42-6x-3y=15[/tex]
[tex]-5x-2y=15-42[/tex]
[tex]-(5x+2y)=-27[/tex]
[tex]5x+2y=27[/tex] ...(v)
Putting [tex]z=14-2x-y[/tex] in (iii), we get
[tex]3x+y+2(14-2x-y)=22[/tex]
[tex]3x+y+28-4x-2y=22[/tex]
[tex]-x-y=22-28[/tex]
[tex]-(x+y)=-6[/tex]
[tex]x+y=6[/tex]
Multiply both sides by 2.
[tex]2x+2y=12[/tex] ..(vi)
Subtracting (vi) from (v), we get
[tex]5x+2y-2x-2y=27-12[/tex]
[tex]3x=15[/tex]
[tex]x=5[/tex]
Putting x=5 in (vi), we get
[tex]2(5)+2y=12[/tex]
[tex]10+2y=12[/tex]
[tex]2y=12-10[/tex]
[tex]2y=2[/tex]
[tex]y=1[/tex]
Putting x=5 and y=1 in (iv), we get
[tex]z=14-2(5)-(1)[/tex]
[tex]z=14-10-1[/tex]
[tex]z=3[/tex]
So, [tex]x=5,y=1,z=3[/tex].
Therefore, the cost of a pizza is $5, cost of a liter of soda is $1 and cost of salad is $3.
