Two white roses and a yellow rose cost 5 pounds. Two white roses and three red roses cost 10.50 pounds. Three yellow roses and two red roses cost 11 pounds. What is the total cost of one red, one white and one yellow rose?

Step-by-step solution

1. 1

   Turn the three sentences into three equations. Let $w$, $y$ and $r$ be the prices in pounds of one white, one yellow and one red rose:

   $2w+y=5,\qquad 2w+3r=10.5,\qquad 3y+2r=11.$

2. 2

   Reduce to one unknown by chaining substitutions. The first equation gives $y$ directly:

   $y=5-2w.$

   Put that into the third equation:

   $3(5-2w)+2r=11\ \Longrightarrow\ 15-6w+2r=11\ \Longrightarrow\ r=3w-2.$

3. 3

   Substitute into the remaining equation. Using $r=3w-2$ in $2w+3r=10.5$:

   $2w+3(3w-2)=10.5\ \Longrightarrow\ 11w-6=10.5\ \Longrightarrow\ w=1.5.$

4. 4

   Back-substitute for the other two prices.

   $y=5-2(1.5)=2,\qquad r=3(1.5)-2=2.5.$

5. 5

   Check all three original conditions. $2(1.5)+2=5$ ✓; $2(1.5)+3(2.5)=3+7.5=10.5$ ✓; $3(2)+2(2.5)=6+5=11$ ✓.

6. 6

   Answer the question actually asked. The total for one of each is

   $w+y+r=1.5+2+2.5=£6.$

   Note that the question only needs the sum, so adding all three equations gives $4w+4y+5r=26.5$ — close to, but not quite, a shortcut; solving the system outright is the reliable route here.