Please explain in detail how to solve the following problem using elimination: It takes a florist 3 h 15 min to make 3 small centerpieces and 3 large centerpieces. It takes

6h 20 min to make 4 small centerpieces and 7 large center pieces. How long does it take to make each small centerpiece and each large centerpiece? Write and solve a system of equations to find your answer.

changing all time values to minutes to avoid fractions,

3S + 3L = 195
4S + 7L = 380

In order to eliminate S or L, we need to manipulate the equations, while not invalidating them.

So, multiply the first by 4 and the second by 3 to get an S coefficient of 12 for both equations:

12S + 12L = 780
12S + 21L = 1140

Now subtract one from the other, and the S's vanish:

9L = 360
L = 40

Now substitute L into either of the two original equations to get

S = 25

So, it takes 25 min. to make a small centerpiece, and 40 min. to make a large one.

To solve this problem using elimination, we can start by writing a system of equations based on the given information.

Let's denote the time it takes to make each small centerpiece as "x" and the time it takes to make each large centerpiece as "y".

From the first statement, we know that it takes 3 hours and 15 minutes (or 3.25 hours) to make 3 small centerpieces and 3 large centerpieces. Therefore, the equation representing this statement is:

3x + 3y = 3.25 (Equation 1)

From the second statement, we know that it takes 6 hours and 20 minutes (or 6.33 hours) to make 4 small centerpieces and 7 large centerpieces. Therefore, the equation representing this statement is:

4x + 7y = 6.33 (Equation 2)

Now, we have a system of equations. To solve it using elimination, we need to eliminate one variable by multiplying one or both of the equations to make the coefficients of one variable the same.

Let's multiply Equation 1 by 4 and Equation 2 by 3 to eliminate x:

(4)(3x + 3y) = (4)(3.25)
(3)(4x + 7y) = (3)(6.33)

This simplifies to:

12x + 12y = 13 (Equation 3)
12x + 21y = 18.99 (Equation 4)

Now, we can subtract Equation 3 from Equation 4 to eliminate x:

(12x + 21y) - (12x + 12y) = 18.99 - 13

This simplifies to:

9y = 5.99

Now, we can solve for y by dividing both sides of the equation by 9:

y = 5.99 / 9

Therefore, y ≈ 0.66.

Now that we have the value of y, we can substitute it back into one of the original equations (Equation 1 or Equation 2) to find the value of x. Let's substitute it into Equation 1:

3x + 3(0.66) = 3.25
3x + 1.98 = 3.25

Now, we can solve for x by subtracting 1.98 from both sides of the equation:

3x = 3.25 - 1.98
3x = 1.27

Finally, we divide both sides of the equation by 3 to solve for x:

x = 1.27 / 3

Therefore, x ≈ 0.42.

Thus, it takes approximately 0.42 hours (or 25 minutes) to make each small centerpiece and approximately 0.66 hours (or 39.6 minutes) to make each large centerpiece.