Question

choose all that correctly interprets the relationship between x and y. each system of equations represents the cost of flowers at a florist.
a white roses cost x dollars and red roses cost y dollars. $2x + y = 22$ and $x + 3y = 26$
thus white roses cost twice as much as red roses
b red roses cost x dollars and daisies cost y dollars. $4x + 4y = 36$ and $x + 2y = 12$
thus red roses cost twice as much as daisies
c tulips cost x dollars and daisies cost y dollars. $2x + 3y = 17$ and $x + y = 7$
thus tulips cost 50 percent more than daisies
d red roses cost x dollars and tulips cost y dollars. $2x + 3y = 24$ and $2x - y = 16$
thus red roses cost 50 percent more than tulips
e white roses cost x dollars and daisies cost y dollars. $3x + 2y = 30$ and $x + y = 11$
thus white roses cost twice as much as daisies

Explanation:

Step1: Solve system for Option A

We have the system:

$$\begin{cases}2x + y = 22 \\ x + 3y = 26\end{cases}$$

Multiply the second equation by 2: $2x + 6y = 52$
Subtract the first equation: $(2x+6y)-(2x+y)=52-22$
$5y=30 \implies y=6$
Substitute $y=6$ into $x+3y=26$: $x+3(6)=26 \implies x=8$
Check if $x=2y$: $8
eq 2(6)=12$. So A is incorrect.

Step2: Solve system for Option B

We have the system:

$$\begin{cases}4x + 4y = 36 \\ x + 2y = 12\end{cases}$$

Simplify first equation: $x + y = 9$
Subtract from second equation: $(x+2y)-(x+y)=12-9$
$y=3$
Substitute $y=3$ into $x+y=9$: $x=6$
Check if $x=2y$: $6=2(3)=6$. So B is correct.

Step3: Solve system for Option C

We have the system:

$$\begin{cases}2x + 3y = 17 \\ x + y = 7\end{cases}$$

Multiply second equation by 2: $2x + 2y = 14$
Subtract from first equation: $(2x+3y)-(2x+2y)=17-14$
$y=4$
Substitute $y=4$ into $x+y=7$: $x=3$
Check if $x=1.5y$: $3
eq 1.5(4)=6$. So C is incorrect.

Step4: Solve system for Option D

We have the system:

$$\begin{cases}2x + 3y = 24 \\ 2x - y = 16\end{cases}$$

Subtract the second equation from the first: $(2x+3y)-(2x-y)=24-16$
$4y=8 \implies y=2$
Substitute $y=2$ into $2x-y=16$: $2x-2=16 \implies 2x=18 \implies x=9$
Check if $x=1.5y$: $9=1.5(2)=3$? No, $9=4.5y$. So D is incorrect.

Step5: Solve system for Option E

We have the system:

$$\begin{cases}3x + 2y = 30 \\ x + y = 11\end{cases}$$

Multiply second equation by 2: $2x + 2y = 22$
Subtract from first equation: $(3x+2y)-(2x+2y)=30-22$
$x=8$
Substitute $x=8$ into $x+y=11$: $y=3$
Check if $x=2y$: $8
eq 2(3)=6$. So E is incorrect.

Answer:

B. Red roses cost x dollars and daisies cost y dollars. $4x + 4y = 36$ and $x + 2y = 12$
Thus, red roses cost twice as much as daisies