Question

modeling real life the diagram shows the prices of two types of ground - cover plants. a gardener can afford to buy 125 vinca plants and 60 phlox plants. (see example 5.)
a. write an equation in standard form that models the possible combinations of vinca and phlox plants the gardener can afford to buy.
b. graph the equation from part (a).
c. find four possible combinations.
modeling real life one bus ride costs $1. a monthly pass for unlimited bus and subway rides costs $36. one subway ride costs $1. a monthly pass for unlimited subway rides costs $36.
a. write an equation in standard form that models the possible combinations of bus and subway rides with the same total cost as the pass.
b. graph the equation from part (a).
c. you ride the bus 60 times in one month. how many times must you ride the subway for the total cost of the rides to equal the cost of the pass? explain your reasoning.

Explanation:

Problem 29

Step1: Define variables

Let $x$ be the number of vinca plants and $y$ be the number of phlox plants. The cost of vinca plants is $\$2.50$ each and phlox plants is $\$2.25$ each, and the gardener can afford 125 vinca plants and 60 phlox plants in total. The equation in standard form $Ax + By=C$ is $2.5x+2.25y = 2.5\times125 + 2.25\times60$. Simplify the right - hand side: $2.5\times125=312.5$ and $2.25\times60 = 135$, so the equation is $2.5x+2.25y=447.5$. Multiply through by 4 to get rid of the decimals: $10x + 9y=1790$.

Step2: Graph the equation

To graph $10x + 9y=1790$, rewrite it in slope - intercept form $y=mx + b$. Solve for $y$: $9y=-10x + 1790$, so $y=-\frac{10}{9}x+\frac{1790}{9}\approx-\frac{10}{9}x + 198.89$. Find the $x$ - intercept by setting $y = 0$: $10x=1790$, $x = 179$. Find the $y$ - intercept by setting $x = 0$: $9y=1790$, $y=\frac{1790}{9}\approx198.89$.

Step3: Find combinations

We need non - negative integer solutions for $x$ and $y$ that satisfy $10x + 9y=1790$.
Let $x = 0$, then $y=\frac{1790}{9}\approx198.89$ (not an integer).
Let $y = 0$, then $x = 179$.
If $x=170$, then $10\times170+9y=1790$, $9y=1790 - 1700=90$, $y = 10$.
If $x = 80$, then $10\times80+9y=1790$, $9y=1790 - 800 = 990$, $y = 110$.
If $x=9$, then $10\times9+9y=1790$, $9y=1790 - 90=1700$, $y=\frac{1700}{9}\approx188.89$ (not an integer).
If $x = 101$, then $10\times101+9y=1790$, $9y=1790 - 1010 = 780$, $y=\frac{780}{9}\approx86.67$ (not an integer).
Four possible combinations: $(x = 179,y = 0)$; $(x=170,y = 10)$; $(x = 80,y=110)$; $(x=0,y=\frac{1790}{9}\approx198.89)$ (we can take non - negative integer solutions like $(179,0),(170,10),(80,110),(0,198)$ (rounding down $y$ value for non - negative integer case)).

Step1: Define variables

Let $x$ be the number of bus rides and $y$ be the number of subway rides. The cost of a bus ride is $\$0.75$ and a subway ride is $\$1$. The monthly pass for unlimited bus and 36 subway rides costs the same as the total cost of individual rides. The equation in standard form is $0.75x + 1y=0.75\times60+1\times36$. Calculate the right - hand side: $0.75\times60 = 45$ and $1\times36=36$, so the equation is $0.75x + y=81$. Multiply through by 4 to get rid of the decimal: $3x+4y = 324$.

Step2: Graph the equation

Rewrite $3x + 4y=324$ in slope - intercept form $y=mx + b$. Solve for $y$: $4y=-3x + 324$, so $y=-\frac{3}{4}x+81$. The $x$ - intercept is found by setting $y = 0$: $3x=324$, $x = 108$. The $y$ - intercept is found by setting $x = 0$: $y = 81$.

Step3: Find combinations

We want non - negative integer solutions for $x$ and $y$ such that $3x + 4y=324$.
If $x = 0$, then $y = 81$.
If $y = 0$, then $x = 108$.
If $x=4$, then $3\times4+4y=324$, $12 + 4y=324$, $4y=312$, $y = 78$.
If $x = 8$, then $3\times8+4y=324$, $24+4y=324$, $4y=300$, $y = 75$.

Step4: Solve for when costs are equal

We want to find when the cost of individual rides equals the cost of the pass. The cost of the pass is based on 60 bus rides and 36 subway rides costing $0.75\times60+1\times36=45 + 36=81$.
Let's find when $0.75x+1y=81$. If we consider the equation $3x + 4y=324$, we know that the cost of the pass is fixed at $\$81$.

Answer:

a. The equation is $10x + 9y=1790$.
b. Graph has $x$ - intercept at $(179,0)$ and $y$ - intercept at $(0,\frac{1790}{9}\approx198.89)$ with slope $m =-\frac{10}{9}$.
c. Four possible combinations: $(179,0),(170,10),(80,110),(0,198)$

Problem 30