Question

Marie is cleaning her clothes out of the closet in her spare bedroom. She started with 650 of her items in the closet. She has been able to continually decrease the number of her items in the closet at a rate of one-half per month. Her brother, Dustin, has started putting 5 items of his clothing in the closet each month as Marie cleans it out. Marie wants to know how many months it will take before the number of items Dustin adds will be equal to the number of her items in the closet.

Create a system of equations to model the situation above, and use it to determine if there are any solutions. If there are any solutions, determine if they are viable or not.

A.) There are no solutions.
B.) There is only one solution, and it is viable.
C.) There is only one solution, and it is not viable.
D.) There are two solutions, and only one is viable.

Answer 1

the answer to your question is c

Answer 2

The answer is B. There is only one solution, and it is viable.

Explanation:

Let's look at what the problem tells us:

• Marie is cleaning her clothes out of the closet in her spare bedroom. She started with 650 of her items in the closet. She has been able to continually decrease the number of her items in the closet at a rate of one-half per month.

Following the line equation:

`y(x)=mx+b`

We need to find b and m in the above statement.

From the definition of the line equation, b is the point in where the line crosses the y axis, so if we make y equal to the number of items, and x equal to time (t), then we only need m, since at time=0, items or y=650.

m or the slope is the ratio, in this case is 1/2 and since it is decreasing it is negative, so m=-1/2.

Marie's equation would be:

`m(t)=-(1)/(2)t+650`

• Dustin, has started putting 5 items of his clothing in the closet each month.

Doing the same analysis done for Marie, at time=0, items=0.

and the slope is m=5

Dustin's equation would be:

`d(t)=5t`

Finally, creating the system of equations from Marie's and Dustin's equations we will have:

`\left \{ {{m(t)=-(1)/(2)t+650} \atop {d(t)=5t}} \right.`

We can then solve for t when both equations are equal, like this:

`-(1)/(2)t+650=5t`

But it is easier to graph it and find the point where both lines cross.

If we look at the attached image, in 118 months and about 5 days (118.18), Marie's items will be equal to 590 and so will Dustin's (we never knew the full capacity of the closet, so it is possible!, yet if the closet's capacity is 650 items total, then this will be impossible)