Question

Use either the substitution or elimination method to solve the following word problem. Your answer should include two equations and the solution to the two equations: Ed and his little sister are saving up money to buy a joint birthday present for their mother. Ed already owes a friend $1, and plans to save $6 per week from his allowance. His sister has $2.66 saved so far and will also save $6 per week from her allowance. The two siblings will soon have saved the same amount towards their mother's gift. How much will each one have saved? How long will that take?

Answer 1

Explanation:

Represent Ed's savings with e and his sister's by s.

Then the function representing Ed's savings is

e(t) = -$1 + ($6/wk)t, where -$1 is the amount he owes and ($6/wk)t represents the amount he will receive from savings.

The function representing his sister's savings is

s(t) = $2.66 + ($6/wk)t

If we equate these two equations and solve the resulting system for e and s, we'll get:

e(t) = -$1 + ($6/wk)t = s(t) = $2.66 + ($6/wk)t

Then $2.66 + ($6/wk)t = -$1 + ($6/wk)t

Combining like terms results in:

$3.66 = ($6/wk)t, which we solve for t:

$3.66

t = --------------- = 0.61 week, which is equivalent to approx. 4 1/2 days.

($6/wk)t

The siblings will have the same amount saved after 4 1/2 days, and that amount will be approximately $2.66 + ($6/wk)(0.61 wk) = $6.32

Answer 2

Let w represent the number of weeks passed.

The amount of money Ed saved is 6w - 1 (I'm subtracting the one dollar that he owes to someone)

His sister saved 6w + 2.66 (I'm adding the money she already saved).

What value of w makes both expressions equal?
We need to solve the equation 6w - 1 = 6w + 2.66

Let's solve it.
If we subtract 6w from both sides, we get -1 = 2.66
So, there is no solution. They will never have saved the same amount.
This makes sense because they started out with different amounts of money saved, but they are saving at the same rate.