At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $\$3.15$. Gary orders a bagel and four muffins, which comes out to $\$7.85$. Larry orders a piece of toast, two bagels, and three muffins, which comes out to $\$9.75$. How many cents does one bagel cost?

Answer 1

Let $\boldsymbol{t}$ denote the cost of one slice of toast, in cents, $\boldsymbol{b}$ denote the cost of one bagel, in cents, and $\boldsymbol{m}$ denote the cost of one muffin, in cents. From the given information, we have the system of equations \begin{align*}
2t+b&=315, \\
b+4m&=785, \\
t+2b+3m&=975.
\end{align*}We desire the value of $\boldsymbol{b}$. Multiplying the first equation by 2 (to make the $t$ terms cancel when summed with the third equation) yields \begin{align*}
4t+2b&=630, \\
t+2b+3m&=975,
\end{align*}and subtracting these two equations gives $3t=345$, or $t=115$. Substituting into the third equation yields $115+2b+3m=975$, or $2b+3m=860$. Multiplying this by 2 and subtracting it from the second equation gives $b+4m-2b-6m=785-2(2b+3m)$, or $-b-2m=-535$. Adding this to 4 times the first equation gives \begin{align*}
(2t+b)+4\left(t+2b+3m\right)&=315+4\left(975\right), \\
6t+9b+12m&=3875, \\
2t+3b+4m&=1291.
\end{align*}Finally, multiplying this equation by $-1$ gives $-2t-3b-4m=-1291$. Summing this with $-b-2m=-535$ yields $-2t-4b-6m=-1826$, or $4b+6m=1826$. Substituting $2b+3m=860$ into this equation yields $4(860)=1826$, so the solution to the system is $b=\boxed{430}$.