Solved on Mar 11, 2024

Determine the length of Teresa's workout plans A and B given the number of clients and total training hours on Friday and Saturday. Plan A: $\square$ hour(s) Plan B: $\square$ hour(s)

STEP 1

Assumptions
1. On Friday, 5 clients did Plan A and 3 clients did Plan B.
2. On Saturday, 7 clients did Plan A and 9 clients did Plan B.
3. Teresa trained her Friday clients for a total of 6 hours.
4. Teresa trained her Saturday clients for a total of 12 hours.
5. Each client does only one plan, either Plan A or Plan B.

STEP 2

Let's denote the length of each Plan A workout as $x$ hours and the length of each Plan B workout as $y$ hours.

STEP 3

Using the information given for Friday, we can set up the first equation based on the total hours Teresa trained her clients:
$5x + 3y = 6$

STEP 4

Using the information given for Saturday, we can set up the second equation based on the total hours Teresa trained her clients:
$7x + 9y = 12$

STEP 5

We now have a system of two linear equations with two variables:
\begin{align} 5x + 3y &= 6 \quad \text{(Equation 1)} \\ 7x + 9y &= 12 \quad \text{(Equation 2)} \end{align}

STEP 6

To solve this system of equations, we can use either the substitution method or the elimination method. Let's use the elimination method to find the values of $x$ and $y$ .

STEP 7

First, we'll try to eliminate one of the variables by making the coefficients of either $x$ or $y$ the same in both equations. Let's eliminate $y$ by multiplying Equation 1 by 3 and Equation 2 by 1.

STEP 8

Multiply Equation 1 by 3:
$3(5x + 3y) = 3(6)$

STEP 9

Multiply Equation 2 by 1 (which doesn't change it):
$1(7x + 9y) = 1(12)$

STEP 10

Write down the new system of equations after multiplication:
\begin{align} 15x + 9y &= 18 \quad \text{(Equation 3)} \\ 7x + 9y &= 12 \quad \text{(Equation 4)} \end{align}

STEP 11

Now, subtract Equation 4 from Equation 3 to eliminate $y$ :
$15x + 9y - (7x + 9y) = 18 - 12$

STEP 12

Perform the subtraction:
$15x - 7x + 9y - 9y = 18 - 12$

STEP 13

Simplify the equation:
$8x = 6$

STEP 14

Divide both sides by 8 to solve for $x$ :
$x = \frac{6}{8}$

STEP 15

Simplify the fraction:
$x = \frac{3}{4}$

STEP 16

Now that we have the value of $x$ , we can substitute it back into one of the original equations to solve for $y$ . Let's use Equation 1:
$5x + 3y = 6$

STEP 17

Substitute $x = \frac{3}{4}$ into Equation 1:
$5\left(\frac{3}{4}\right) + 3y = 6$

STEP 18

Multiply 5 by $\frac{3}{4}$ :
$\frac{15}{4} + 3y = 6$

STEP 19

Convert 6 to a fraction with the same denominator as $\frac{15}{4}$ to simplify the equation:
$\frac{15}{4} + 3y = \frac{24}{4}$

STEP 20

Subtract $\frac{15}{4}$ from both sides to solve for $3y$ :
$3y = \frac{24}{4} - \frac{15}{4}$

STEP 21

Subtract the fractions:
$3y = \frac{9}{4}$

STEP 22

Divide both sides by 3 to solve for $y$ :
$y = \frac{9}{4} \div 3$

STEP 23

Simplify the division:
$y = \frac{9}{4} \times \frac{1}{3}$

STEP 24

Multiply the fractions:
$y = \frac{9}{12}$

STEP 25

Simplify the fraction:
$y = \frac{3}{4}$

STEP 26

We have found that both Plan A and Plan B workouts last $\frac{3}{4}$ hour, which is equivalent to 45 minutes.
Length of each Plan A workout: $\frac{3}{4}$ hour(s) Length of each Plan B workout: $\frac{3}{4}$ hour(s)

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Determine the length of Teresa's workout plans A and B given the number of clients and total training hours on Friday and Saturday. Plan A: □\square□ hour(s) Plan B: □\square□ hour(s)

STEP 1

STEP 2

Let's denote the length of each Plan A workout as xxx hours and the length of each Plan B workout as yyy hours.

STEP 3

Using the information given for Friday, we can set up the first equation based on the total hours Teresa trained her clients:5x+3y=65x + 3y = 65x+3y=6

STEP 4

Using the information given for Saturday, we can set up the second equation based on the total hours Teresa trained her clients:7x+9y=127x + 9y = 127x+9y=12

STEP 5

We now have a system of two linear equations with two variables:\begin{align*} 5x + 3y &= 6 \quad \text{(Equation 1)} \\ 7x + 9y &= 12 \quad \text{(Equation 2)} \end{align*}

STEP 6

To solve this system of equations, we can use either the substitution method or the elimination method. Let's use the elimination method to find the values of xxx and yyy.

STEP 7

First, we'll try to eliminate one of the variables by making the coefficients of either xxx or yyy the same in both equations. Let's eliminate yyy by multiplying Equation 1 by 3 and Equation 2 by 1.

STEP 8

Multiply Equation 1 by 3:3(5x+3y)=3(6)3(5x + 3y) = 3(6)3(5x+3y)=3(6)

STEP 9

Multiply Equation 2 by 1 (which doesn't change it):1(7x+9y)=1(12)1(7x + 9y) = 1(12)1(7x+9y)=1(12)

STEP 10

Write down the new system of equations after multiplication:\begin{align*} 15x + 9y &= 18 \quad \text{(Equation 3)} \\ 7x + 9y &= 12 \quad \text{(Equation 4)} \end{align*}

STEP 11

Now, subtract Equation 4 from Equation 3 to eliminate yyy:15x+9y−(7x+9y)=18−1215x + 9y - (7x + 9y) = 18 - 1215x+9y−(7x+9y)=18−12

STEP 12

Perform the subtraction:15x−7x+9y−9y=18−1215x - 7x + 9y - 9y = 18 - 1215x−7x+9y−9y=18−12

STEP 13

Simplify the equation:8x=68x = 68x=6

STEP 14

Divide both sides by 8 to solve for xxx:x=68x = \frac{6}{8}x=86​

STEP 15

Simplify the fraction:x=34x = \frac{3}{4}x=43​

STEP 16

Now that we have the value of xxx, we can substitute it back into one of the original equations to solve for yyy. Let's use Equation 1:5x+3y=65x + 3y = 65x+3y=6

STEP 17

Substitute x=34x = \frac{3}{4}x=43​ into Equation 1:5(34)+3y=65\left(\frac{3}{4}\right) + 3y = 65(43​)+3y=6

STEP 18

Multiply 5 by 34\frac{3}{4}43​:154+3y=6\frac{15}{4} + 3y = 6415​+3y=6

STEP 19

Convert 6 to a fraction with the same denominator as 154\frac{15}{4}415​ to simplify the equation:154+3y=244\frac{15}{4} + 3y = \frac{24}{4}415​+3y=424​

STEP 20

Subtract 154\frac{15}{4}415​ from both sides to solve for 3y3y3y:3y=244−1543y = \frac{24}{4} - \frac{15}{4}3y=424​−415​

STEP 21

Subtract the fractions:3y=943y = \frac{9}{4}3y=49​

STEP 22

Divide both sides by 3 to solve for yyy:y=94÷3y = \frac{9}{4} \div 3y=49​÷3

STEP 23

Simplify the division:y=94×13y = \frac{9}{4} \times \frac{1}{3}y=49​×31​

STEP 24

Multiply the fractions:y=912y = \frac{9}{12}y=129​

STEP 25

Simplify the fraction:y=34y = \frac{3}{4}y=43​

STEP 26

We have found that both Plan A and Plan B workouts last 34\frac{3}{4}43​ hour, which is equivalent to 45 minutes.Length of each Plan A workout: 34\frac{3}{4}43​ hour(s) Length of each Plan B workout: 34\frac{3}{4}43​ hour(s)

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Determine the length of Teresa's workout plans A and B given the number of clients and total training hours on Friday and Saturday. Plan A: $\square$ hour(s) Plan B: $\square$ hour(s)

Let's denote the length of each Plan A workout as $x$ hours and the length of each Plan B workout as $y$ hours.

Using the information given for Friday, we can set up the first equation based on the total hours Teresa trained her clients:
$5x + 3y = 6$

Using the information given for Saturday, we can set up the second equation based on the total hours Teresa trained her clients:
$7x + 9y = 12$

We now have a system of two linear equations with two variables:
\begin{align} 5x + 3y &= 6 \quad \text{(Equation 1)} \\ 7x + 9y &= 12 \quad \text{(Equation 2)} \end{align}

To solve this system of equations, we can use either the substitution method or the elimination method. Let's use the elimination method to find the values of $x$ and $y$ .

First, we'll try to eliminate one of the variables by making the coefficients of either $x$ or $y$ the same in both equations. Let's eliminate $y$ by multiplying Equation 1 by 3 and Equation 2 by 1.

Multiply Equation 1 by 3:
$3(5x + 3y) = 3(6)$

Multiply Equation 2 by 1 (which doesn't change it):
$1(7x + 9y) = 1(12)$

Write down the new system of equations after multiplication:
\begin{align} 15x + 9y &= 18 \quad \text{(Equation 3)} \\ 7x + 9y &= 12 \quad \text{(Equation 4)} \end{align}

Now, subtract Equation 4 from Equation 3 to eliminate $y$ :
$15x + 9y - (7x + 9y) = 18 - 12$

Perform the subtraction:
$15x - 7x + 9y - 9y = 18 - 12$

Simplify the equation:
$8x = 6$

Divide both sides by 8 to solve for $x$ :
$x = \frac{6}{8}$

Simplify the fraction:
$x = \frac{3}{4}$

Now that we have the value of $x$ , we can substitute it back into one of the original equations to solve for $y$ . Let's use Equation 1:
$5x + 3y = 6$

Substitute $x = \frac{3}{4}$ into Equation 1:
$5\left(\frac{3}{4}\right) + 3y = 6$

Multiply 5 by $\frac{3}{4}$ :
$\frac{15}{4} + 3y = 6$

Convert 6 to a fraction with the same denominator as $\frac{15}{4}$ to simplify the equation:
$\frac{15}{4} + 3y = \frac{24}{4}$

Subtract $\frac{15}{4}$ from both sides to solve for $3y$ :
$3y = \frac{24}{4} - \frac{15}{4}$

Subtract the fractions:
$3y = \frac{9}{4}$

Divide both sides by 3 to solve for $y$ :
$y = \frac{9}{4} \div 3$

Simplify the division:
$y = \frac{9}{4} \times \frac{1}{3}$

Multiply the fractions:
$y = \frac{9}{12}$

Simplify the fraction:
$y = \frac{3}{4}$

We have found that both Plan A and Plan B workouts last $\frac{3}{4}$ hour, which is equivalent to 45 minutes.
Length of each Plan A workout: $\frac{3}{4}$ hour(s) Length of each Plan B workout: $\frac{3}{4}$ hour(s)