Problem A$-6($Alg$)$

Logan Manufacturing wants to mix two fuels, A and $B,$ for its trucks to minimize cost. It needs no fewer than $3,100$ gallons to run its trucks during the next month. It has a maximum fuel storage capacity of $3,700$ gallons. There are $2,300$ gallons of fuel A and $4,200$ gallons of fuel $B$ available. The mixed fuel must have an octane rating of no less than $77.$

When fuels are mixed, the amount of fuel obtained is just equal to the sum of the amounts put in$.$ The octane rating is the weighted average of the individual octanes, weighted in proportion to the respective volumes.

The following is known: Fuel A has an octane of $84$ and costs $$0\mathrm{.}90$ per gallon. Fuel $B$ has an octane of $60$ and costs $$1\mathrm{.}20$ per gallon.

a$.$ Write the equations expressing this information. $($Leave no cells blank $-$ be certain to enter $"0"$ wherever required. Negative answers should be indicated by a minus sign. Round your answers to $2$ decimal places.$)$