QuestionTalent show with 16 solo acts ( $x$ min) and 3 ensemble acts ( $y$ min). First show lasts 151 min, second 95 min with 8 best solo acts. Write a system of equations to model the situation.
$16x + 3y = 151$ $8x + 3y = 95$
Solve the system.
(x, y)

Studdy Solution

STEP 1

Assumptions
1. There are 16 solo acts and 3 ensemble acts in the first show.
2. The first show lasts 151 minutes in total.
3. The 8 best solo acts will perform again in the second show.
4. The second show also includes the 3 ensemble acts and lasts 95 minutes in total.
5. Each solo act lasts $x$ minutes, and each ensemble act lasts $y$ minutes.
6. We need to find the correct system of equations to model the situation.

STEP 2

We need to write an equation for the total duration of the first show. The total time is the sum of the time for all solo acts and all ensemble acts.
$16x + 3y = 151$
This equation represents the total duration of the first show.

STEP 3

Next, we write an equation for the total duration of the second show. Again, the total time is the sum of the time for the 8 best solo acts and all ensemble acts.
$8x + 3y = 95$
This equation represents the total duration of the second show.

STEP 4

Now we have a system of two equations with two variables ( $x$ and $y$ ) which models the situation:
\begin{align} 16x + 3y &= 151 \quad \text{(Equation 1)}\\ 8x + 3y &= 95 \quad \text{(Equation 2)} \end{align}

STEP 5

To solve this system of equations, we can use the method of elimination or substitution. We will use the elimination method by subtracting Equation 2 from Equation 1 to eliminate $y$ .

STEP 6

Subtract Equation 2 from Equation 1:
\begin{align} (16x + 3y) - (8x + 3y) &= 151 - 95\\ 16x - 8x + 3y - 3y &= 56\\ 8x &= 56 \end{align}

STEP 7

Now, solve for $x$ by dividing both sides of the equation by 8:
\begin{align} 8x &= 56\\ x &= \frac{56}{8}\\ x &= 7 \end{align}
So, each solo act lasts 7 minutes.

STEP 8

Now that we have the value of $x$ , we can substitute it back into either Equation 1 or Equation 2 to find $y$ . We will use Equation 2 for convenience.

STEP 9

Substitute $x = 7$ into Equation 2:
\begin{align} 8(7) + 3y &= 95\\ 56 + 3y &= 95 \end{align}

STEP 10

Subtract 56 from both sides of the equation to solve for $y$ :
\begin{align} 56 + 3y - 56 &= 95 - 56\\ 3y &= 39 \end{align}

STEP 11

Now, solve for $y$ by dividing both sides of the equation by 3:
\begin{align} 3y &= 39\\ y &= \frac{39}{3}\\ y &= 13 \end{align}
So, each ensemble act lasts 13 minutes.
The solution to the system of equations is $x = 7$ and $y = 13$ , which means each solo act lasts 7 minutes and each ensemble act lasts 13 minutes.
The ordered pair is $(7, 13)$ .

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QuestionTalent show with 16 solo acts ( $x$ min) and 3 ensemble acts ( $y$ min). First show lasts 151 min, second 95 min with 8 best solo acts. Write a system of equations to model the situation.
$16x + 3y = 151$ $8x + 3y = 95$
Solve the system.
(x, y)

Studdy Solution

STEP 1

STEP 2

We need to write an equation for the total duration of the first show. The total time is the sum of the time for all solo acts and all ensemble acts.
$16x + 3y = 151$
This equation represents the total duration of the first show.

STEP 3

Next, we write an equation for the total duration of the second show. Again, the total time is the sum of the time for the 8 best solo acts and all ensemble acts.
$8x + 3y = 95$
This equation represents the total duration of the second show.

STEP 4

Now we have a system of two equations with two variables ( $x$ and $y$ ) which models the situation:
\begin{align} 16x + 3y &= 151 \quad \text{(Equation 1)}\\ 8x + 3y &= 95 \quad \text{(Equation 2)} \end{align}

STEP 5

To solve this system of equations, we can use the method of elimination or substitution. We will use the elimination method by subtracting Equation 2 from Equation 1 to eliminate $y$ .

STEP 6

Subtract Equation 2 from Equation 1:
\begin{align} (16x + 3y) - (8x + 3y) &= 151 - 95\\ 16x - 8x + 3y - 3y &= 56\\ 8x &= 56 \end{align}

STEP 7

Now, solve for $x$ by dividing both sides of the equation by 8:
\begin{align} 8x &= 56\\ x &= \frac{56}{8}\\ x &= 7 \end{align}
So, each solo act lasts 7 minutes.

STEP 8

Now that we have the value of $x$ , we can substitute it back into either Equation 1 or Equation 2 to find $y$ . We will use Equation 2 for convenience.

STEP 9

Substitute $x = 7$ into Equation 2:
\begin{align} 8(7) + 3y &= 95\\ 56 + 3y &= 95 \end{align}

STEP 10

Subtract 56 from both sides of the equation to solve for $y$ :
\begin{align} 56 + 3y - 56 &= 95 - 56\\ 3y &= 39 \end{align}

STEP 11

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QuestionTalent show with 16 solo acts (xxx min) and 3 ensemble acts (yyy min). First show lasts 151 min, second 95 min with 8 best solo acts. Write a system of equations to model the situation.16x+3y=15116x + 3y = 15116x+3y=151 8x+3y=958x + 3y = 958x+3y=95Solve the system.(x, y)

Studdy Solution

STEP 1

STEP 2

We need to write an equation for the total duration of the first show. The total time is the sum of the time for all solo acts and all ensemble acts.16x+3y=15116x + 3y = 15116x+3y=151This equation represents the total duration of the first show.

STEP 3

Next, we write an equation for the total duration of the second show. Again, the total time is the sum of the time for the 8 best solo acts and all ensemble acts.8x+3y=958x + 3y = 958x+3y=95This equation represents the total duration of the second show.

STEP 4

Now we have a system of two equations with two variables (xxx and yyy) which models the situation:\begin{align*} 16x + 3y &= 151 \quad \text{(Equation 1)}\\ 8x + 3y &= 95 \quad \text{(Equation 2)} \end{align*}

STEP 5

To solve this system of equations, we can use the method of elimination or substitution. We will use the elimination method by subtracting Equation 2 from Equation 1 to eliminate yyy.

STEP 6

Subtract Equation 2 from Equation 1:\begin{align*} (16x + 3y) - (8x + 3y) &= 151 - 95\\ 16x - 8x + 3y - 3y &= 56\\ 8x &= 56 \end{align*}

STEP 7

Now, solve for xxx by dividing both sides of the equation by 8:\begin{align*} 8x &= 56\\ x &= \frac{56}{8}\\ x &= 7 \end{align*}So, each solo act lasts 7 minutes.

STEP 8

Now that we have the value of xxx, we can substitute it back into either Equation 1 or Equation 2 to find yyy. We will use Equation 2 for convenience.

STEP 9

Substitute x=7x = 7x=7 into Equation 2:\begin{align*} 8(7) + 3y &= 95\\ 56 + 3y &= 95 \end{align*}

STEP 10

Subtract 56 from both sides of the equation to solve for yyy:\begin{align*} 56 + 3y - 56 &= 95 - 56\\ 3y &= 39 \end{align*}

STEP 11

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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.

QuestionTalent show with 16 solo acts ( $x$ min) and 3 ensemble acts ( $y$ min). First show lasts 151 min, second 95 min with 8 best solo acts. Write a system of equations to model the situation.
$16x + 3y = 151$ $8x + 3y = 95$
Solve the system.
(x, y)

We need to write an equation for the total duration of the first show. The total time is the sum of the time for all solo acts and all ensemble acts.
$16x + 3y = 151$
This equation represents the total duration of the first show.

Next, we write an equation for the total duration of the second show. Again, the total time is the sum of the time for the 8 best solo acts and all ensemble acts.
$8x + 3y = 95$
This equation represents the total duration of the second show.

Now we have a system of two equations with two variables ( $x$ and $y$ ) which models the situation:
\begin{align} 16x + 3y &= 151 \quad \text{(Equation 1)}\\ 8x + 3y &= 95 \quad \text{(Equation 2)} \end{align}

To solve this system of equations, we can use the method of elimination or substitution. We will use the elimination method by subtracting Equation 2 from Equation 1 to eliminate $y$ .

Subtract Equation 2 from Equation 1:
\begin{align} (16x + 3y) - (8x + 3y) &= 151 - 95\\ 16x - 8x + 3y - 3y &= 56\\ 8x &= 56 \end{align}

Now, solve for $x$ by dividing both sides of the equation by 8:
\begin{align} 8x &= 56\\ x &= \frac{56}{8}\\ x &= 7 \end{align}
So, each solo act lasts 7 minutes.

Now that we have the value of $x$ , we can substitute it back into either Equation 1 or Equation 2 to find $y$ . We will use Equation 2 for convenience.

Substitute $x = 7$ into Equation 2:
\begin{align} 8(7) + 3y &= 95\\ 56 + 3y &= 95 \end{align}

Subtract 56 from both sides of the equation to solve for $y$ :
\begin{align} 56 + 3y - 56 &= 95 - 56\\ 3y &= 39 \end{align}