Solved on Jan 23, 2024

Describe a system of equations to solve for how long it takes two worker groups painting a bridge at different rates to paint the same amount, and how much each group will have painted.
$x$ = days until equal painting, $y$ = km painted by each group Group 1: 3 km + 2 km/day * $x$ days = $y$ km Group 2: 1 km + 4 km/day * $x$ days = $y$ km

STEP 1

Assumptions
1. The first group has already painted 3 kilometers and paints 2 kilometers per day.
2. The second group has already painted 1 kilometer and paints 4 kilometers per day.
3. Let $x$ represent the number of days after which both groups will have painted the same amount of the bridge.
4. The total distance painted by each group will be equal when they have painted the same amount.

STEP 2

We need to write two equations that represent the total distance painted by each group after $x$ days.
For the first group: $Total\,distance\,painted\,by\,first\,group = Initial\,distance + (Rate\,of\,painting \times Number\,of\,days)$
For the second group: $Total\,distance\,painted\,by\,second\,group = Initial\,distance + (Rate\,of\,painting \times Number\,of\,days)$

STEP 3

Now, we write the equations using the given information.
For the first group: $Total\,distance\,painted\,by\,first\,group = 3 + 2x$
For the second group: $Total\,distance\,painted\,by\,second\,group = 1 + 4x$

STEP 4

Since both groups will have painted the same amount of the bridge after $x$ days, we can set the two expressions equal to each other.
$3 + 2x = 1 + 4x$

STEP 5

Now we solve for $x$ .
Subtract $2x$ from both sides of the equation to get the $x$ terms on one side: $3 + 2x - 2x = 1 + 4x - 2x$ $3 = 1 + 2x$

STEP 6

Subtract 1 from both sides of the equation to isolate the term with $x$ : $3 - 1 = 1 + 2x - 1$ $2 = 2x$

STEP 7

Divide both sides of the equation by 2 to solve for $x$ : $\frac{2}{2} = \frac{2x}{2}$ $x = 1$

STEP 8

Now that we have the value of $x$ , we can find out how much of the bridge each group will have painted after $x$ days.
For the first group: $Total\,distance\,painted\,by\,first\,group = 3 + 2x$ $Total\,distance\,painted\,by\,first\,group = 3 + 2(1)$ $Total\,distance\,painted\,by\,first\,group = 3 + 2$ $Total\,distance\,painted\,by\,first\,group = 5$

STEP 9

For the second group: $Total\,distance\,painted\,by\,second\,group = 1 + 4x$ $Total\,distance\,painted\,by\,second\,group = 1 + 4(1)$ $Total\,distance\,painted\,by\,second\,group = 1 + 4$ $Total\,distance\,painted\,by\,second\,group = 5$

STEP 10

Therefore, it will take 1 day for both groups of workers to have painted 5 kilometers of the bridge each.
In days, both groups of workers will have painted 5 kilometers of the bridge.

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STEP 1

STEP 2

STEP 3

Now, we write the equations using the given information.
For the first group: $Total\,distance\,painted\,by\,first\,group = 3 + 2x$
For the second group: $Total\,distance\,painted\,by\,second\,group = 1 + 4x$

STEP 4

Since both groups will have painted the same amount of the bridge after $x$ days, we can set the two expressions equal to each other.
$3 + 2x = 1 + 4x$

STEP 5

Now we solve for $x$ .
Subtract $2x$ from both sides of the equation to get the $x$ terms on one side: $3 + 2x - 2x = 1 + 4x - 2x$ $3 = 1 + 2x$

STEP 6

Subtract 1 from both sides of the equation to isolate the term with $x$ : $3 - 1 = 1 + 2x - 1$ $2 = 2x$

STEP 7

Divide both sides of the equation by 2 to solve for $x$ : $\frac{2}{2} = \frac{2x}{2}$ $x = 1$

STEP 8

STEP 9

For the second group: $Total\,distance\,painted\,by\,second\,group = 1 + 4x$ $Total\,distance\,painted\,by\,second\,group = 1 + 4(1)$ $Total\,distance\,painted\,by\,second\,group = 1 + 4$ $Total\,distance\,painted\,by\,second\,group = 5$

STEP 10

Therefore, it will take 1 day for both groups of workers to have painted 5 kilometers of the bridge each.
In days, both groups of workers will have painted 5 kilometers of the bridge.

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STEP 1

STEP 2

STEP 3

STEP 4

Since both groups will have painted the same amount of the bridge after x x x days, we can set the two expressions equal to each other.3+2x=1+4x 3 + 2x = 1 + 4x 3+2x=1+4x

STEP 5

Now we solve for x x x.Subtract 2x 2x 2x from both sides of the equation to get the x x x terms on one side: 3+2x−2x=1+4x−2x 3 + 2x - 2x = 1 + 4x - 2x 3+2x−2x=1+4x−2x 3=1+2x 3 = 1 + 2x 3=1+2x

STEP 6

Subtract 1 from both sides of the equation to isolate the term with x x x: 3−1=1+2x−1 3 - 1 = 1 + 2x - 1 3−1=1+2x−1 2=2x 2 = 2x 2=2x

STEP 7

Divide both sides of the equation by 2 to solve for x x x: 22=2x2 \frac{2}{2} = \frac{2x}{2} 22​=22x​ x=1 x = 1 x=1

STEP 8

STEP 9

STEP 10

Therefore, it will take 1 day for both groups of workers to have painted 5 kilometers of the bridge each.In days, both groups of workers will have painted 5 kilometers of the bridge.

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

Since both groups will have painted the same amount of the bridge after $x$ days, we can set the two expressions equal to each other.
$3 + 2x = 1 + 4x$

Now we solve for $x$ .
Subtract $2x$ from both sides of the equation to get the $x$ terms on one side: $3 + 2x - 2x = 1 + 4x - 2x$ $3 = 1 + 2x$

Subtract 1 from both sides of the equation to isolate the term with $x$ : $3 - 1 = 1 + 2x - 1$ $2 = 2x$

Divide both sides of the equation by 2 to solve for $x$ : $\frac{2}{2} = \frac{2x}{2}$ $x = 1$

Therefore, it will take 1 day for both groups of workers to have painted 5 kilometers of the bridge each.
In days, both groups of workers will have painted 5 kilometers of the bridge.