Math / Algebra

QuestionAlgebra $2>$ E. 7 Solve a system of equations using substitution: w You have prizes to reveal! Go to your game boar Video
Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. Danny is going to hire a makeup artist for a fashion show and is comparing prices. Elizabeth charges $\$ 42$ as a booking fee and an additional $\$ 98$ per hour. Gary charges $\$ 52$ per hour, plus a booking fee of $\$ 88$ . Depending on the length of the show, the cost could end up being the same for either artist. How long would the show be? What would the cost be?
If the show lasted for $\square$ hours, the cost would always be \\square$ .

Studdy Solution

STEP 1

1. Elizabeth charges a booking fee of \$42 and \$98 per hour.
2. Gary charges a booking fee of \$88 and \$52 per hour.
3. We want to find the number of hours for which the total cost of hiring either artist is the same.
4. We need to determine the cost at that number of hours.

STEP 2

1. Define variables for the problem.
2. Write equations for the total cost for each artist.
3. Set the equations equal to each other and solve for the variable using substitution.
4. Calculate the cost using the found value of the variable.

STEP 3

Define variables for the problem.
Let $h$ represent the number of hours the show lasts.

STEP 4

Write equations for the total cost for each artist.
For Elizabeth, the total cost $C_E$ is given by: $C_E = 42 + 98h$
For Gary, the total cost $C_G$ is given by: $C_G = 88 + 52h$

STEP 5

Set the equations equal to each other and solve for the variable using substitution.
Since we want the costs to be the same, set $C_E = C_G$ : $42 + 98h = 88 + 52h$
Subtract $52h$ from both sides: $42 + 46h = 88$
Subtract 42 from both sides: $46h = 46$
Divide both sides by 46: $h = 1$

STEP 6

Calculate the cost using the found value of the variable.
Substitute $h = 1$ back into either equation to find the cost.
Using Elizabeth's cost equation: $C_E = 42 + 98(1) = 42 + 98 = 140$
Thus, if the show lasted for $\boxed{1}$ hour, the cost would always be \$ $\boxed{140}$.

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Studdy Solution

STEP 1

1. Elizabeth charges a booking fee of \$42 and \$98 per hour.
2. Gary charges a booking fee of \$88 and \$52 per hour.
3. We want to find the number of hours for which the total cost of hiring either artist is the same.
4. We need to determine the cost at that number of hours.

STEP 2

1. Define variables for the problem.
2. Write equations for the total cost for each artist.
3. Set the equations equal to each other and solve for the variable using substitution.
4. Calculate the cost using the found value of the variable.

STEP 3

Define variables for the problem.
Let $h$ represent the number of hours the show lasts.

STEP 4

Write equations for the total cost for each artist.
For Elizabeth, the total cost $C_E$ is given by: $C_E = 42 + 98h$
For Gary, the total cost $C_G$ is given by: $C_G = 88 + 52h$

STEP 5

Set the equations equal to each other and solve for the variable using substitution.
Since we want the costs to be the same, set $C_E = C_G$ : $42 + 98h = 88 + 52h$
Subtract $52h$ from both sides: $42 + 46h = 88$
Subtract 42 from both sides: $46h = 46$
Divide both sides by 46: $h = 1$

STEP 6

Calculate the cost using the found value of the variable.
Substitute $h = 1$ back into either equation to find the cost.
Using Elizabeth's cost equation: $C_E = 42 + 98(1) = 42 + 98 = 140$
Thus, if the show lasted for $\boxed{1}$ hour, the cost would always be \$ \(\boxed{140}\).

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Studdy Solution

STEP 1

1. Elizabeth charges a booking fee of \$42 and \$98 per hour.2. Gary charges a booking fee of \$88 and \$52 per hour.3. We want to find the number of hours for which the total cost of hiring either artist is the same.4. We need to determine the cost at that number of hours.

STEP 2

1. Define variables for the problem.2. Write equations for the total cost for each artist.3. Set the equations equal to each other and solve for the variable using substitution.4. Calculate the cost using the found value of the variable.

STEP 3

Define variables for the problem.Let h h h represent the number of hours the show lasts.

STEP 4

Write equations for the total cost for each artist.For Elizabeth, the total cost CE C_E CE​ is given by: CE=42+98h C_E = 42 + 98h CE​=42+98hFor Gary, the total cost CG C_G CG​ is given by: CG=88+52h C_G = 88 + 52h CG​=88+52h

STEP 5

STEP 6

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Elizabeth charges a booking fee of \$42 and \$98 per hour.
2. Gary charges a booking fee of \$88 and \$52 per hour.
3. We want to find the number of hours for which the total cost of hiring either artist is the same.
4. We need to determine the cost at that number of hours.

1. Define variables for the problem.
2. Write equations for the total cost for each artist.
3. Set the equations equal to each other and solve for the variable using substitution.
4. Calculate the cost using the found value of the variable.

Define variables for the problem.
Let $h$ represent the number of hours the show lasts.

Write equations for the total cost for each artist.
For Elizabeth, the total cost $C_E$ is given by: $C_E = 42 + 98h$
For Gary, the total cost $C_G$ is given by: $C_G = 88 + 52h$