Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Franco is going to send some flowers to his wife. Westford Florist charges $\$1$ per rose, plus $\$20$ for the vase. Nate's Flowers, in contrast, charges $\$3$ per rose and $\$18$ for the vase. If Franco orders the bouquet with a certain number of roses, the cost will be the same with either flower shop. How many roses would there be? What would the total cost be? $\newline$ If the bouquet contains $\_\_\_\_$ roses, it will cost $\$\_\_\_\_$ .

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Solve a system of equations using substitution: word problems

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Franco is going to send some flowers to his wife. Westford Florist charges

\$1

per rose, plus

\$20

for the vase. Nate's Flowers, in contrast, charges

\$3

per rose and

\$18

for the vase. If Franco orders the bouquet with a certain number of roses, the cost will be the same with either flower shop. How many roses would there be? What would the total cost be?

\newline

If the bouquet contains

\_\_\_\_

roses, it will cost

\$\_\_\_\_

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of roses in the bouquet. $\newline$ Let $y$ be the total cost of the bouquet.
Write Westford Florist Equation: We can write the equation for Westford Florist as follows: $\newline$ y = $\$1$ * $x$ + $\$20$ (Cost per rose * Number of roses + Cost of vase) $\newline$ So, the equation is $y = x + 20$ .
Write Nate's Flowers Equation: We can write the equation for Nate's Flowers as follows: $\newline$ y = $\$3$ * x + $\$18$ (Cost per rose * Number of roses + Cost of vase) $\newline$ So, the equation is $y = 3x + 18$ .
Set Equations Equal: Now we have two equations: $\newline$ $1$ ) $y = x + 20$ (Westford Florist) $\newline$ $2$ ) $y = 3x + 18$ (Nate's Flowers) $\newline$ Since the total cost $y$ is the same for both, we can set the equations equal to each other to solve for $x$ . $\newline$ $x + 20 = 3x + 18$
Subtract $x$ : Subtract $x$ from both sides to start solving for $x$ : $x + 20 - x = 3x + 18 - x$ $20 = 2x + 18$
Subtract $18$ : Subtract $18$ from both sides to isolate the term with $x$ : $\newline$ $20 - 18 = 2x + 18 - 18$ $\newline$ $2 = 2x$
Divide by $2$ : Divide both sides by $2$ to solve for x: $\newline$ $\frac{2}{2} = \frac{2x}{2}$ $\newline$ $1 = x$
Find Total Cost: Now that we have the number of roses $x$ , we can find the total cost $y$ using either of the original equations. Let's use the equation from Westford Florist: $\newline$ $y = x + 20$ $\newline$ $y = 1 + 20$ $\newline$ $y = \$(21)$
Final Result: We have found that the bouquet contains $1$ rose and it will cost $\$21$ at both Westford Florist and Nate's Flowers.