[High School - Math] Linear Systems

Relambrien · 01-22-2008, 09:17 PM

I was helping my Algebra I brother with his math homework tonight. One of the problems he had to do went something like this:

There are two rental car companies. One charges $17 per day of rental and $0.13 per mile driven. Another charges $19 per day of rental and $0.07 per mile driven. How many miles must you drive for the costs to be equal?

I looked at this and thought "Well there can't be any one solution, as the number of miles necessary will fluctuate based on the number of days the car is rented." Every other problem in the packet had a single solution, so I was curious about this.

So the first thing I did was to create two equations to model the situation, and set them equal to each other:

17x + 0.13y = 19x + 0.07y

Simplifying, this becomes:

0.13y = 2x + 0.07y

Plugging in values of 1 and 2 for x, the number of necessary miles came about to be 33.3... and 66.6... respectively (I could also have extended the solution of the equation to model this, but just plugging in values was simpler). So the number of miles necessary to create equal cost is 33.3...x, where x is number of days of rental.

Another blank asked for total costs of car rental when this number of miles was driven (so it would be the same for both car companies). By setting 33.3... for y and 1 for x in both of the original equations, the total cost came out to be 21.3...x, where x is number of days of rental.

I'm fairly positive that this is correct, but I'm wary, as that was the only problem in the entire packet which required a general, rather than a particular, solution. I made a note of this on my brother's packet, so that the teacher would know. I'm wondering if this is correct, however. I can't help but think that I missed something obvious when the solution to this one problem is so drastically different than the solutions to the others.

If anyone could help confirm or deny this, I'd appreciate it. Thanks.

mexadiaz · 01-22-2008, 09:52 PM

You've done the problem correctly but I believe it only wants the answer for a one-day rental.

Relambrien · 01-22-2008, 10:03 PM

See, that's what I had originally thought, but the exact wording made it seem otherwise...what I have written in the first post is merely a paraphrase from my memory.

But it's good to know that at least the calculations were correct.

**Tasselfoot** · 01-22-2008, 11:18 PM

yea. your equation and process is right... but for a basic algebra question, it would seem that they'd give you the number of days of the rental, to make the only unknown that of the miles driven. otherwise you're left with what you did... having to make your answer contingent on a variable still. which seems complex for algebra I.

lord_carbo · 01-23-2008, 05:00 PM

Quote:

Originally Posted by Relambrien

Simplifying, this becomes:

0.13y = 2x + 0.07y

Nope. You have a y term on each side and you didn't divide. Subtract the one from the right. This is the most simplified form:

0.13y = 2x + 0.07y

(subtract)

0.06y = 2x

(divide)

0.03y = x

Relambrien · 01-23-2008, 06:34 PM

Most simplified, yes, but I wasn't going to bother simplifying it fully when what I had worked just fine. I usually just get things as simplified as they need to be for me to be comfortable working with them, which I got at 0.13y = 2x + 0.07y.

In other words, simplifying it fully just wasn't necessary.

01-22-2008, 09:17 PM	#1
Relambrien FFR Player Join Date: Dec 2006 Location: Pittsburgh, PA Age: 32 Posts: 1,644	[High School - Math] Linear Systems I was helping my Algebra I brother with his math homework tonight. One of the problems he had to do went something like this: There are two rental car companies. One charges $17 per day of rental and $0.13 per mile driven. Another charges $19 per day of rental and $0.07 per mile driven. How many miles must you drive for the costs to be equal? I looked at this and thought "Well there can't be any one solution, as the number of miles necessary will fluctuate based on the number of days the car is rented." Every other problem in the packet had a single solution, so I was curious about this. So the first thing I did was to create two equations to model the situation, and set them equal to each other: 17x + 0.13y = 19x + 0.07y Simplifying, this becomes: 0.13y = 2x + 0.07y Plugging in values of 1 and 2 for x, the number of necessary miles came about to be 33.3... and 66.6... respectively (I could also have extended the solution of the equation to model this, but just plugging in values was simpler). So the number of miles necessary to create equal cost is 33.3...x, where x is number of days of rental. Another blank asked for total costs of car rental when this number of miles was driven (so it would be the same for both car companies). By setting 33.3... for y and 1 for x in both of the original equations, the total cost came out to be 21.3...x, where x is number of days of rental. I'm fairly positive that this is correct, but I'm wary, as that was the only problem in the entire packet which required a general, rather than a particular, solution. I made a note of this on my brother's packet, so that the teacher would know. I'm wondering if this is correct, however. I can't help but think that I missed something obvious when the solution to this one problem is so drastically different than the solutions to the others. If anyone could help confirm or deny this, I'd appreciate it. Thanks. Last edited by Relambrien; 01-22-2008 at 09:20 PM..

01-22-2008, 09:52 PM	#2
mexadiaz FFR Player Join Date: May 2004 Posts: 29	Re: [High School - Linear Systems] You've done the problem correctly but I believe it only wants the answer for a one-day rental.

01-22-2008, 10:03 PM	#3
Relambrien FFR Player Join Date: Dec 2006 Location: Pittsburgh, PA Age: 32 Posts: 1,644	Re: [High School - Linear Systems] See, that's what I had originally thought, but the exact wording made it seem otherwise...what I have written in the first post is merely a paraphrase from my memory. But it's good to know that at least the calculations were correct.

01-22-2008, 11:18 PM	#4
Tasselfoot Retired BOSS Join Date: Jul 2003 Location: Widget Heaven Age: 40 Posts: 25,184	Re: [High School - Linear Systems] yea. your equation and process is right... but for a basic algebra question, it would seem that they'd give you the number of days of the rental, to make the only unknown that of the miles driven. otherwise you're left with what you did... having to make your answer contingent on a variable still. which seems complex for algebra I. __________________ RIP

01-23-2008, 06:34 PM	#6
Relambrien FFR Player Join Date: Dec 2006 Location: Pittsburgh, PA Age: 32 Posts: 1,644	Re: [High School - Math] Linear Systems Most simplified, yes, but I wasn't going to bother simplifying it fully when what I had worked just fine. I usually just get things as simplified as they need to be for me to be comfortable working with them, which I got at 0.13y = 2x + 0.07y. In other words, simplifying it fully just wasn't necessary. Last edited by Relambrien; 01-23-2008 at 06:37 PM..

Currently Active Users Viewing This Thread: 1 (0 members and 1 guests)