$1.$In planning her retirement, Liza deposits some money at $3\%$ interest, with twice as much deposited at $3\mathrm{.}5\%.$ Find the amount deposited at each rate if the total annual interest income is $$1300.$

She deposited $ at $3\%$ and $ at $3\mathrm{.}5\%.$

$2.$Membership warehouse clubs offer shoppers low prices, along with rewards of cash back on club purchases. If the yearly fee for a warehouse club membersnip is $$100$ and the reward rate is $4\%$ on club purchases for the year, then the linear equation $y=100-0\mathrm{.}04x$ models the actual yearly cost of the membership y$,$ in dollars. Here $x$ represents the yearly amount of club purchases, also in dollars.

a$)$ Determine the actual yearly cost of the membership if club purchases for the year are $$1600.$

b$)$ What amount of club purchases would reduce the actual yearly cost of the membership to $$34?$

c$)$ How much would a member have to spend in yearly club purchases to reduce the yearly membership cost to $$0?$

a$)$ The actual yearly cost of the membership if club purchases for the year are $$1600$ is $ $($Simplify your answer. Type an integer or a decimal.$)$