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Quiz 22
1.) There is a concert for which VIP tickets cost $\$20$and regular tickets cost$\$10$. A total of $75$ tickets were sold and $\$1050$was earned. How many tickets of each type were sold? Solution: Let$x$represent the number of$\$20$ tickets and let $y$ represent the number of $\$10$tickets. We are told$75$total tickets are sold which gives us the equation $$x+y=75.$$ The total income we earn by selling the$\$20$ tickets is $20x$ and the total income we earn by selling the $\$10$tickets is$10y$. To say that we earned$\$1050$ gives us the equation $$20x+10y=1050.$$ Therefore we have the system of equations $$\left\{ \begin{array}{lll} x+y &= 75 & \quad (i) \\ 20x+10y&= 1050 & \quad (ii). \end{array} \right.$$ Solve $(i)$ for $y$ to get $y=75-x$. Plug this into $(ii)$ to get $$20x + 10(75-x) = 1050.$$ Distribute the $10$ to get $$20x + 750 - 10x = 1050.$$ Combine like terms on the left and subtract $750$ to get $$10x = 300.$$ Divide by $10$ to get $$x = 30.$$ To find $y$, plug this value of $x$ into one of the equations -- plugging into equation $(i)$ is probably easier: $$30+y=75.$$ Subtract $30$ to find $$y = 45.$$ Therefore there were $30$ of the $\$20$tickets sold and$45$of the$\$10$ tickets sold.

2.) You want to mix Colombian coffee worth $\$10$per pound and Turkish coffee worth$\$14$ per pound. How much of each should you mix together to get $30$ pounds of a coffee blend worth $\$12$a pound? Solution: Let$x$represent the number of pounds of Colombian coffee and let$y$represent the number of pounds of Turkish coffee. To say we want$30$pounds of coffee in the end means we want $$x+y=30.$$ The price of$x$pounds of Colombian coffee is$10x$and the price of$y$pounds of Turkish coffee is$14y$. To say we want the mixture of$x+y$pounds to be worth$\$12$ means that $12(x+y)$. This means the other equation is $$10x+14y=12(x+y).$$ (NOTE: Since we already knew that $x+y=30$ you could also write that equationintead as $10x+14y=(12)(30)$ or $10x+14y=360$)
Therefore we have the system $$\left\{ \begin{array}{lll} x+y &= 30 & \quad (i) \\ 10x+14y &= 12(x+y) & \quad (ii). \end{array} \right.$$ We solve $(ii)$ for $x$: distribute the $12$ to get $$10x+14y=12x+12y.$$ Subtract both sides by $10x$ and subtract both sides by $12y$ to get $$2y = 2x.$$ Therefore $$y=x.$$ Plug this into $(i)$ to get $$x+x=30,$$ or $$2x=30,$$ or $$x=15.$$ Now find the value of $y$ and get $y=15$. Therefore if we mix $15$ pounds of Colombian coffee with $15$ pounds of Turkish coffee, we get $30$ pounds of a blend worth $\$12$a pound. 3.) Graph $$y \leq x+2.$$ Solution: First we will draw the line$y=x+2$as a solid line: We will use the test point$(0,0)$and plug it into the original inequality: $$0 \leq 0 + 2$$ yielding $$0 \leq 2,$$ which is true. Therefore we shade the half of the plane containing that tst point: 4.) Graph $$y-1 > 2x.$$ Solution: First we will graph the line$y-1=2x$with a dotted line (we don't have$\geq$so we use dotted line). To easily see what line this is solve for$y$to get$y=2x+1$. Draw this: We will use the test point$(0,0)\$ and plug it into the original inequality: $$0-1 > 2(0).$$ Simplifying this yields $$-1 > 0$$ which is false. Therefore we shade the half of the plane not containing the test point: