Solved on Feb 02, 2024

David has completed 2 oil changes and can do 1 every 2 hours. Ezra can do 3 oil changes per hour. Find the number of oil changes each has completed after a given time.
Let $x$ be the time in hours and $y_1$ and $y_2$ be the number of oil changes completed by David and Ezra, respectively.
$y_1 = 2 + \frac{x}{2}$ $y_2 = 3x$

STEP 1

Assumptions
1. David has already completed 2 oil changes.
2. David can complete 1 oil change every 2 hours.
3. Ezra can complete 3 oil changes every hour.
4. We are looking to find out after how many hours will they have completed the same number of oil changes.

STEP 2

Let's define variables for the system of equations.
- Let $x$ be the number of hours since the competition started. - Let $D(x)$ be the total number of oil changes David has completed after $x$ hours. - Let $E(x)$ be the total number of oil changes Ezra has completed after $x$ hours.

STEP 3

Write the equation for David's total oil changes.
Since David has already completed 2 oil changes and completes 1 additional oil change every 2 hours, his total oil changes after $x$ hours can be represented as:
$D(x) = 2 + \frac{1}{2}x$

STEP 4

Write the equation for Ezra's total oil changes.
Since Ezra can complete 3 oil changes every hour, his total oil changes after $x$ hours can be represented as:
$E(x) = 3x$

STEP 5

We now have a system of equations that represents the total number of oil changes completed by David and Ezra after $x$ hours.
$\begin{cases} D(x) = 2 + \frac{1}{2}x \\ E(x) = 3x \end{cases}$

STEP 6

To find out after how many hours they will have completed the same number of oil changes, we need to find the value of $x$ for which $D(x) = E(x)$ .

STEP 7

Set the two equations equal to each other to find the point of intersection.
$2 + \frac{1}{2}x = 3x$

STEP 8

Solve for $x$ by first moving all terms involving $x$ to one side of the equation.
$\frac{1}{2}x - 3x = -2$

STEP 9

Combine like terms.
$-\frac{5}{2}x = -2$

STEP 10

Divide both sides by $-\frac{5}{2}$ to solve for $x$ .
$x = \frac{-2}{-\frac{5}{2}}$

STEP 11

Simplify the fraction by multiplying the numerator and denominator by the reciprocal of the denominator.
$x = \frac{-2}{-1} \times \frac{2}{5}$

STEP 12

Calculate the value of $x$ .
$x = 2 \times \frac{2}{5} = \frac{4}{5}$

STEP 13

Convert the fraction to a decimal to find the number of hours.
$x = \frac{4}{5} = 0.8$

STEP 14

Now that we have the value of $x$ , we can find the number of oil changes each has completed after 0.8 hours.
For David: $D(0.8) = 2 + \frac{1}{2}(0.8)$

STEP 15

Calculate David's total oil changes after 0.8 hours.
$D(0.8) = 2 + \frac{1}{2}(0.8) = 2 + 0.4 = 2.4$

STEP 16

For Ezra: $E(0.8) = 3(0.8)$

STEP 17

Calculate Ezra's total oil changes after 0.8 hours.
$E(0.8) = 3(0.8) = 2.4$

STEP 18

David and Ezra will both have completed 2.4 oil changes after 0.8 hours. Since they cannot complete a fraction of an oil change, we round this number to the nearest whole number.

STEP 19

Round the number of oil changes to the nearest whole number.
$\text{Number of oil changes completed by both} = \lceil 2.4 \rceil = 3$

STEP 20

Therefore, after 0.8 hours, both David and Ezra will have completed 3 oil changes each.
The solution to the system of equations is $x = 0.8$ hours, with each having completed 3 oil changes.

Was this helpful?

STEP 1

Assumptions1. David has already completed 2 oil changes.2. David can complete 1 oil change every 2 hours.3. Ezra can complete 3 oil changes every hour.4. We are looking to find out after how many hours will they have completed the same number of oil changes.

STEP 2

STEP 3

Write the equation for David's total oil changes.Since David has already completed 2 oil changes and completes 1 additional oil change every 2 hours, his total oil changes after x x x hours can be represented as:D(x)=2+12x D(x) = 2 + \frac{1}{2}x D(x)=2+21​x

STEP 4

Write the equation for Ezra's total oil changes.Since Ezra can complete 3 oil changes every hour, his total oil changes after x x x hours can be represented as:E(x)=3x E(x) = 3x E(x)=3x

STEP 5

We now have a system of equations that represents the total number of oil changes completed by David and Ezra after x x x hours.{D(x)=2+12xE(x)=3x \begin{cases} D(x) = 2 + \frac{1}{2}x \\ E(x) = 3x \end{cases} {D(x)=2+21​xE(x)=3x​

STEP 6

To find out after how many hours they will have completed the same number of oil changes, we need to find the value of x x x for which D(x)=E(x) D(x) = E(x) D(x)=E(x).

STEP 7

Set the two equations equal to each other to find the point of intersection.2+12x=3x 2 + \frac{1}{2}x = 3x 2+21​x=3x

STEP 8

Solve for x x x by first moving all terms involving x x x to one side of the equation.12x−3x=−2 \frac{1}{2}x - 3x = -2 21​x−3x=−2

STEP 9

Combine like terms.−52x=−2 -\frac{5}{2}x = -2 −25​x=−2

STEP 10

Divide both sides by −52 -\frac{5}{2} −25​ to solve for x x x.x=−2−52 x = \frac{-2}{-\frac{5}{2}} x=−25​−2​

STEP 11

Simplify the fraction by multiplying the numerator and denominator by the reciprocal of the denominator.x=−2−1×25 x = \frac{-2}{-1} \times \frac{2}{5} x=−1−2​×52​

STEP 12

Calculate the value of x x x.x=2×25=45 x = 2 \times \frac{2}{5} = \frac{4}{5} x=2×52​=54​

STEP 13

Convert the fraction to a decimal to find the number of hours.x=45=0.8 x = \frac{4}{5} = 0.8 x=54​=0.8

STEP 14

Now that we have the value of x x x, we can find the number of oil changes each has completed after 0.8 hours.For David: D(0.8)=2+12(0.8) D(0.8) = 2 + \frac{1}{2}(0.8) D(0.8)=2+21​(0.8)

STEP 15

Calculate David's total oil changes after 0.8 hours.D(0.8)=2+12(0.8)=2+0.4=2.4 D(0.8) = 2 + \frac{1}{2}(0.8) = 2 + 0.4 = 2.4 D(0.8)=2+21​(0.8)=2+0.4=2.4

STEP 16

For Ezra: E(0.8)=3(0.8) E(0.8) = 3(0.8) E(0.8)=3(0.8)

STEP 17

Calculate Ezra's total oil changes after 0.8 hours.E(0.8)=3(0.8)=2.4 E(0.8) = 3(0.8) = 2.4 E(0.8)=3(0.8)=2.4

STEP 18

David and Ezra will both have completed 2.4 oil changes after 0.8 hours. Since they cannot complete a fraction of an oil change, we round this number to the nearest whole number.

STEP 19

Round the number of oil changes to the nearest whole number.Number of oil changes completed by both=⌈2.4⌉=3 \text{Number of oil changes completed by both} = \lceil 2.4 \rceil = 3 Number of oil changes completed by both=⌈2.4⌉=3

STEP 20

Therefore, after 0.8 hours, both David and Ezra will have completed 3 oil changes each.The solution to the system of equations is x=0.8 x = 0.8 x=0.8 hours, with each having completed 3 oil changes.

Start learning now

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

Assumptions
1. David has already completed 2 oil changes.
2. David can complete 1 oil change every 2 hours.
3. Ezra can complete 3 oil changes every hour.
4. We are looking to find out after how many hours will they have completed the same number of oil changes.

Write the equation for David's total oil changes.
Since David has already completed 2 oil changes and completes 1 additional oil change every 2 hours, his total oil changes after $x$ hours can be represented as:
$D(x) = 2 + \frac{1}{2}x$

Write the equation for Ezra's total oil changes.
Since Ezra can complete 3 oil changes every hour, his total oil changes after $x$ hours can be represented as:
$E(x) = 3x$

We now have a system of equations that represents the total number of oil changes completed by David and Ezra after $x$ hours.
$\begin{cases} D(x) = 2 + \frac{1}{2}x \\ E(x) = 3x \end{cases}$

To find out after how many hours they will have completed the same number of oil changes, we need to find the value of $x$ for which $D(x) = E(x)$ .

Set the two equations equal to each other to find the point of intersection.
$2 + \frac{1}{2}x = 3x$

Solve for $x$ by first moving all terms involving $x$ to one side of the equation.
$\frac{1}{2}x - 3x = -2$

Combine like terms.
$-\frac{5}{2}x = -2$

Divide both sides by $-\frac{5}{2}$ to solve for $x$ .
$x = \frac{-2}{-\frac{5}{2}}$

Simplify the fraction by multiplying the numerator and denominator by the reciprocal of the denominator.
$x = \frac{-2}{-1} \times \frac{2}{5}$

Calculate the value of $x$ .
$x = 2 \times \frac{2}{5} = \frac{4}{5}$

Convert the fraction to a decimal to find the number of hours.
$x = \frac{4}{5} = 0.8$

Now that we have the value of $x$ , we can find the number of oil changes each has completed after 0.8 hours.
For David: $D(0.8) = 2 + \frac{1}{2}(0.8)$

Calculate David's total oil changes after 0.8 hours.
$D(0.8) = 2 + \frac{1}{2}(0.8) = 2 + 0.4 = 2.4$

For Ezra: $E(0.8) = 3(0.8)$

Calculate Ezra's total oil changes after 0.8 hours.
$E(0.8) = 3(0.8) = 2.4$

Round the number of oil changes to the nearest whole number.
$\text{Number of oil changes completed by both} = \lceil 2.4 \rceil = 3$

Therefore, after 0.8 hours, both David and Ezra will have completed 3 oil changes each.
The solution to the system of equations is $x = 0.8$ hours, with each having completed 3 oil changes.