Emmanuel added 8 links per minute to his chain mail. Allesia started 20 minutes after Emmanuel and added 13 links per minute to her chain mail. How long had Emmanuel worked when Allesia caught up to him, and how many links had he added?

Emmanuel had worked for 52 minutes and had added 416 links.

(Khan Academy)

Step-by-step explanation:

Let's use t to represent the number of minutes Emmanuel had worked and l to represent the number of links added to the chain mail.

We have two unknown values, so we can create and solve a system of two equations that represent this context.

Modeling the context:

Emmanuel added 8 links a minute, so for Emmanuel:

l=8t

Allesia worked for 20 minutes less than Emmanuel did, so she worked t-20. Allesia added 13 links a minute, so for Allesia:

l=13 (t-20)

Now we have the following system of equations to represent our situation:

l=8t

l=13 (t-20)

Solving for t by substituting

Let's substitute the value of l from the second equation into the first equation.

l=8t

13 (t-20) = 8t

13t-260=8t

5t-260=0

5t=260

t=52

Allesia catches up to Emmanuel after Emmanuel had worked 52 minutes.

Solving for l by substituting

We have one unknown left. Let's substitute the value we got for t back into one of the equations to solve for l.

l=8t

l=8 (52)

l=416

Emmanuel had added 416 links.

Emmanuel had worked for 52 minutes and had added 416 links.

Answer: 52 minutes, 416 links

Step-by-step explanation:

8*20=160

13-8=5

160/5=32

32+20=52

52*8=416