Use the substitution method to solve the system of equations.
Let k be the cost of a knife and s be the cost of a spoon.
Since a knife is four times the cost of a spoon, then:
[tex]k=4s[/tex]
Since 12 spoons and 16 knives cost 105.64, then:
[tex]12s+16k=105.64[/tex]
Replace the expression for k in terms of s into the second equation:
[tex]12s+16(4s)=105.64[/tex]
We obtained an equation in a single variable. Solve for s:
[tex]\begin{gathered} \Rightarrow12s+64s=105.64 \\ \Rightarrow76s=105.64 \\ \Rightarrow s=\frac{105.64}{76} \\ \therefore s=1.39 \end{gathered}[/tex]
Replace the value of s into the expression for k to find the cost of a knife:
[tex]\begin{gathered} k=4s \\ \Rightarrow k=4(1.39) \\ \therefore k=5.56 \end{gathered}[/tex]
Therefore, the cost of one knife is equal to £5.56.
