Systems of two equations -- word problems

[Shadow367]

I'm having some trouble understanding a few concepts of solving systems of two equations. I can do it just fine when I already have the equations, but word problems are giving me some trouble. Here's three examples of problems I'm not sure how to do:

Solve:
1. Casella's Catering is planning a wedding reception. The bride and groom would like to serve a nut mixture containing 25% peanuts. Casella has available mixtures that are either 40% peanuts or 10% peanuts. How much of each type should be mixed to get a 20-lb. mixture that is 25% peanuts?
I thought that you would start by setting up two equations:
x+y=20
0.40x+0.10y=0.25
However, when I tried to work it out after that, I couldn't seem to get it right. I don't know if I'm setting up the equations incorrectly or what, and I'm not really sure what to fix so that I can work it out after I get the equations. The only other thing I could think of was multiplying by 100 to get rid of the decimals, but I didn't know if I should do that or not.

2. The octane rating of gasoline is a measure of the amount of isooctane in the gas. The 2002 Dodge Neon RT requires 91-octane gasoline. How much 87-octane gas and 93-octane gas should Kasey mix in order to make 12 gallons of 91-octane gas for her Neon RT?
So far I could only work out this:
x+y=12
and I'm not even sure if that's correct. I wasn't sure about the other equation, because 87x+93y=91 didn't make sense to me.

3. A picture frame measures 14 cm by 20 cm, and 84 cm[sup:2e5hhyyg]2[/sup:2e5hhyyg] of picture shows. Find the width of the frame.
The only thing I could think of to start this problem was finding the area of the picture frame, which would be 14*20=280 cm[sup:2e5hhyyg]2[/sup:2e5hhyyg], then subtracting the area of the picture showing (84 cm[sup:2e5hhyyg]2[/sup:2e5hhyyg]) from the area of the frame (280 cm[sup:2e5hhyyg]2[/sup:2e5hhyyg]). I wasn't sure if that's what I needed to do, or if I was starting it completely wrong, and I'm not sure where to go next.

I've tried these problems over and over and just can't seem to figure them out. Can anyone help me with the steps to solve these? I want to practice these kinds of problems, I'm just not sure how to do them and trying different ones many times seems like it isn't helping

 

[mmm4444bot]

x + y = 20

0.40x + 0.10y = 0.25

This is a good start. There's only two issues to fix.

The first issue: Please write down your symbol definitions before using your symbols to form equations!

(The meaning of symbol x and of symbol y is fairly obvious; yet, it's good form to always define symbols with explicit statements at the very beginning.)

x = the pounds of 40%-peanut mix used

y = the pounds of 10%-peanut mix used

The equation x + y = 20 models the statement, "the pounds of 40%-peanut mix used PLUS the pounds of 10%-peanut mix used YIELDS a batch of nuts that weighs 20 pounds". This is good.

The second issue: The equation 0.4x + 0.1y = 0.25 models the statement, "40% of x PLUS 10% of y EQUALS 25%".

25% of what?

You need to write the equation so that it models, "40% of x PLUS 10% of y EQUALS 25% of x and y combined".

Can you do that?


The only other thing I could think of was multiplying by 100 to get rid of the decimals, but I didn't know if I should do that or not.

It's not necessary (especially if you're using a calculator to do your arithmetic), but clearing the decimals would give you Whole numbers to work with and it would make Denis very proud.

 

[mmm4444bot]

2. The octane rating of gasoline is a measure of the amount of isooctane in the gas. The 2002 Dodge Neon RT requires 91-octane gasoline. How much 87-octane gas and 93-octane gas should Kasey mix in order to make 12 gallons of 91-octane gas for her Neon RT?

x + y = 12

87x + 93y = 91

Same two issues.

Define your symbols.

The 91 should be a factor of (x + y).

 

[mmm4444bot]

3. A picture frame measures 14 cm by 20 cm, and 84 cm[sup:1cb6akhy]2[/sup:1cb6akhy] of picture shows. Find the width of the frame.

This is sloppy wording. They don't want the width of the frame; they want the width of the frame's uniform border around the picture.

Did you draw and label a diagram?

There's an easier setup for finding the width of the border than a system of two equations.

Let x = the width of the border

From your diagram, it should be clear that the width of the picture itself is (14 - 2x) centimeters.

Likewise, the height of the picture itself is (20 - 2x) centimeters.

You know the formula for the area of the image rectangle (in terms of the two dimensions above), and you know the area itself.

Put it all together, to form a single equation to solve for x.

 

[Shadow367]

Re:

x + y = 20

0.40x + 0.10y = 0.25

This is a good start. There's only two issues to fix.

The first issue: Please write down your symbol definitions before using your symbols to form equations!

(The meaning of symbol x and of symbol y is fairly obvious; yet, it's good form to always define symbols with explicit statements at the very beginning.)

x = the pounds of 40%-peanut mix used

y = the pounds of 10%-peanut mix used

The equation x + y = 20 models the statement, "the pounds of 40%-peanut mix used PLUS the pounds of 10%-peanut mix used YIELDS a batch of nuts that weighs 20 pounds". This is good.

The second issue: The equation 0.4x + 0.1y = 0.25 models the statement, "40% of x PLUS 10% of y EQUALS 25%".

25% of what?

You need to write the equation so that it models, "40% of x PLUS 10% of y EQUALS 25% of x and y combined".

Can you do that?


The only other thing I could think of was multiplying by 100 to get rid of the decimals, but I didn't know if I should do that or not.

It's not necessary (especially if you're using a calculator to do your arithmetic), but clearing the decimals would give you Whole numbers to work with and it would make Denis very proud.

Thank you! Sorry about not defining; we never have to do it in classes I've taken and so I'm not used to writing it down In any case, after reading what you wrote, the equation should be 0.40x+0.20y=0.25t, where t represents the total amount of mixture needed, so t would equal 20 since you need a 20-lb. mixture in total. Am I thinking of this correctly? After multiplying that out (assuming it is the correct equation), it would become 0.40x+0.20y=5 and then I would be able to solve from there.

 

[Shadow367]

Re:

2. The octane rating of gasoline is a measure of the amount of isooctane in the gas. The 2002 Dodge Neon RT requires 91-octane gasoline. How much 87-octane gas and 93-octane gas should Kasey mix in order to make 12 gallons of 91-octane gas for her Neon RT?

x + y = 12

87x + 93y = 91

Same two issues.

Define your symbols.

The 91 should be a factor of (x + y).

So, the symbols defined are:
x = gallons of 87-octane gasoline in the mixture
y = gallons of 93-octane gasoline in the mixture

However, I'm not quite sure what you mean by 91 should be a factor of (x+y). Sorry, we've only really worked with factors in factoring out binomials/trinomials, and I haven't heard my professor say anything similar to that, so I'm a bit confused as to what you mean.

 

[JeffM]

I'm having some trouble understanding a few concepts of solving systems of two equations. I can do it just fine when I already have the equations, but word problems are giving me some trouble. Here's three examples of problems I'm not sure how to do:

Solve:
1. Casella's Catering is planning a wedding reception. The bride and groom would like to serve a nut mixture containing 25% peanuts. Casella has available mixtures that are either 40% peanuts or 10% peanuts. How much of each type should be mixed to get a 20-lb. mixture that is 25% peanuts?
I thought that you would start by setting up two equations:
x+y=20
0.40x+0.10y=0.25
However, when I tried to work it out after that, I couldn't seem to get it right. I don't know if I'm setting up the equations incorrectly or what, and I'm not really sure what to fix so that I can work it out after I get the equations. The only other thing I could think of was multiplying by 100 to get rid of the decimals, but I didn't know if I should do that or not.

I've tried these problems over and over and just can't seem to figure them out. Can anyone help me with the steps to solve these? I want to practice these kinds of problems, I'm just not sure how to do them and trying different ones many times seems like it isn't helping

Let's MAKE SURE you are on the right path for the first problem. We can deal with the remaining ones once you are SURE you understand the first one.

mmm is providing you great advice. Writing down definitions of your terms is vital to getting help here because we are not looking at your book so you must be extra careful in giving us the right information.

MORE IMPORTANT, developing the habit of writing what symbol stands for what unknown will help you immensely in avoiding confusion in your own head. Whenever you say to yourself, "I don't know where to start," start by writing down symbols and definitions for what is unknown. That is a start, and it puts your ideas down into concrete form that you can SEE. Your vision is stronger than your unassisted imagination.

OK

x = weight of the 40% mixture.
y = weight of the 10% mixture.
.4x = the weight of peanuts in the 40% mixture. (40% is a GIVEN.)
.1y = the weight of peanuts in the 10% mixture. (10% is a GIVEN.)

USING YOUR SYMBOLS x AND y, DESCRIBE YOUR 25% MIXTURE.

weight of the 25% mixture = (x + y).
weight of peanuts in the 25% mixture = the weight of peanuts in the 40% mixture + the weight of peanuts in the 10% mixture = (.4x + .1y).

Are you GIVEN any other information?

Yes, the weight of the 25% mixture = 20 pounds.
The weight of the peanuts in 20 pounds of a 25% mixture = .25 * 20 = 5.

So NOW you have symbolic descriptions and numeric descriptions of the same things so you can EQUATE them.

(x + y) = 20.
(.4x + .1y) = 5.

The word-problem has been transformed into a purely symbolic form that you know how to solve.

This step-by-step procedure may look very lengthy. And if the problem seems very easy to you, you may be able to skip steps without error. But whenever you are stuck on a word-problem, this kind of approach will work.

Now let's see you follow the same kinds of steps on your next two problems to make sure you have it.

PS Factor as a NOUN just means one of the things being multiplied in a multiplication.

 

[Denis]

2. The octane rating of gasoline is a measure of the amount of isooctane in the gas. The 2002 Dodge Neon RT requires 91-octane gasoline. How much 87-octane gas and 93-octane gas should Kasey mix in order to make 12 gallons of 91-octane gas for her Neon RT?

I handle these this way; but that's me!

   x @ 87
12-x @ 93
=========
  12 @ 91

[87x + 93(12-x)] / 12 = 91
Solve for x: you'll get x = 4.

So, as general case:

  x @ a
  y @ b
========
x+y @ c

(ax + by) / (x + y) = c