Algebra Linear Equations with word problems?

Question

I understand how to write the equations from the word problem, what I am having problems with is the step -by-step completion of the problem.
Example:

The combined cost of one advance ticket to a show and one same-day ticket was $65. It is known that 40 tickets were sold in advance and 35 the same day, for total receipts of $2425 . What was the price of each kind of ticket?

x+y=40
40x+35y=2425

Now how do I solve this step-by-step? Help please because I know there are rules, I just don't know what they are and guessing what x and y might be seems to take too long :o(. Thanks

Ooops, sorry, I was typing faster then my brain lol. Most of you caught it though, x+y=65 not x+y=40. Sorry about that.

Answer 1

Hi,


Your first equation should be x + y = 65
Take that first equation and solve it for x: x = 65 - y

Substitute the expression "65 - y" into the second equation in place of x. Then solve for y.

40x + 35y = 2425
40(65 - y) + 35y = 2425
2600 - 40y + 35y = 2425
2600 - 5y = 2425
-5y = -175
y = 35, so a same day ticket costs $35

Since x + y = 65 and y = 35, then x = 30
An advance ticket costs $30.

I hope that helps!! :-)

Answer 2

You look like a nice girl who really wants to learn as opposed to just looking for someone to give you the answer so I am going to take my time and help you. First of all, you are absolutely right. Guessing is not even an option because there are literally an infinite amount of answers. You might think that the answers are whole numbers but they don't have to be. There is nothing in the problem indicating that they are whole numbers. What is the prices were something like $9.25 or $6.37? Even if they were whole numbers, your chances at guessing and getting the right solution are almost zero.

Second of all, your equations are not entirely correct but a good habit would be to label your variables so that you know exactly what each one means while you are doing your problem AND what they represent at the end. You don't want to do the whole problem and then forget at the end what x meant to begin with. This first step is called declaration (declaration of variables). So the first two lines in your solution would be

Let x=the number of tickets sold in advance
Let y=the number of tickets sold the same day

In this particular problem, the declaration might look trivial and a waste of time but it is a good habit to get into because if you get a longer, more complicated problem, with 5 variables, you will be glad that you kept track.

Now, your setup is almost correct. The first equation should be equal to 65 not 40. I think this is just a typo. Another thing to remember at this point is that there is a result in upper mathematics (which we use here) that however many variables you have, that is exactly how many distinct equations we need to have a unique solution. Meaning that you are using two variables, you better have two equations and two equations only. Otherwise we are in trouble. You can't have only equation and you can't have four equations. From the entire problem, you must extract two equations. Since you have done that, we can proceed.

Now, since you have two equations with two unknowns (we call it a 2x2 system or a two by two system) there are several methods to solve them. I don't know which ones or how many you have learned so I will do the easiest one with this system. The two equations together are called a system. Usually, you look at the problem and it tells you which method will be better but there is no way for me to tell you which is better and why. This is just something that comes from experience. Just to let you know that all methods work on all systems but certain methods are better with certain systems (because of less work). So in your case, substitution is better than anything else.

First equation gives us that x=65-y. Now we simply plug this first derived equation into the second one (for x) and get
40(65-y)+35y=2425 so now you have one equation with one unknown which we can simplify and solve. This gives us
2600-5y=2425
which gives us 5y=175 which tells us that y must be 35. Using that fact and then going back to the first equation, we find out that x must have been 30.

So advance tickets were $30 and the same day tickets were $35. You can also plug these numbers in BOTH of the equations and see if they hold true. Both and only both of those equations should hold true. One is not enough.

For example, is 35+30=65? Yes it is so the first equation is satisfied. Second, is 40(30)+35(35)=2425? Yes it is so we are good. Checking your answer is another good habit to get into.

Answer 3

The first step, really, is to say what your variables mean!

If A = cost of an advanced ticket, and
S = cost of a same-day ticket, then the equations are:

a) A + S = 65 (not 40) and
b) 40*A + 35*S = 2425.

You want A and S, so solve the system. From a), we have

A = 65-S, note signs, subs this into b)

b) 40*(65-S) + 35*S = 2425, expand

40*65 - 40S + 35S = 2425, gather

-40 S + 35S = 2425 - 40*65 or

-5S = -175, div by -5 or
S = 35, so a same-day costs $ 35, and

A = 65 - 35 = 30 $ for advanced tickets.

Answer 4

OK - let's go throught this:

First of all, you need to see that your first equation is not correct. It should be

x+y=65 (the price of one advance and one day of show)

Now take a look at both equations:
x+y =65
40x+35y=2425;

Your goal is to eliminate one of the variables. You can choose either one. In this case, I am going to elminate y by multiplying the top equation by 35. Here is how it goes:

35x + 35y = 2275
40x + 35y = 2425; ok. Now subtract the bottom from the top.

-5x = -150
x = 30.

So there were 30 advance tickets sold. Now substitute back into the first equations.

30 + y = 65
y =35

there were 35 tickets sold on the same day.

Check in the other equation:

40(30) + 35(35) = 2425??
1200 + 1225 = 2425??
2425 = 2425 YES!!

Hope this helps.

Answer 5

Because x+y = 65, we have x = 65-y

so 40x+35y = 40(65-y)+35y =2600 -40y+35y=2600-5y =2425

so 5y = 2600-2425 = 175 --> y = 175/5 = 35

x = 65-y=65-35 = 30

Answer 6

x+y=65
40x+35y=2425

x=65-y from the first equation .Plug that in for x into the second equation and solve.

2600-40y+35y=2425
-5y=-175
y= $35
then x= $30

Answer 7

x = amount of money for advanced ticket
y = amount of money for same-day ticket
x + y = 65 .... or ... y = 65 - x
40x + 35y = 2425
40x + 35(65 - x) = 2425
40x + 2275 - 35x = 2425
5x = 150
x = 30
Now, y = 65 - x = 65 - 30 = 35

Answer 8

the easiet way to solce this is by substitution one of the three methods substitution, elimination, or graphing

you can either set the top equation equal to x or y
y=-x+40

then substitue for y int he next equation

40x +35(-x +40)=2425
then just work it out
40x -35x +354+2425
then add like terms
5x+354=2425
then subtract 354 form both sides
5x=2071

Answer 9

Look at the first equation...

x+y=40

You can manipulate this by moving x to right of the equal sign.

y=40-x

Now you know what Y is... in term of x. Now, notice there is Y in the second equation. You can substitute it.

40x+35(40-x)=2425

If you simplify this, you'll know what X value is. Once you know it, plug that X value into the first equation to find what Y is.

Answer 10

Maggie has a job working in an office for $10 an hour. Maggie earns by working in the office = $10x assuming she worked for x hours Maggie has another job driving a tractor for $12 an hour. Maggie earns by driving the tractor = $12y assuming she worked for y hours. Maggie works at both jobs and earns $480 in one week. 10x + 12y = 480