Question

Calvin bought 5 packets of cheese and 3 burgers for $29.99. At the same prices, Alex bought 3 packets of cheese and 7 burgers for $32.71. Find the price of one packet of cheese by writing a system of equations, then solve using Cramer's Rule.

Answer 1

Let \(x\) be the price of a packet of cheese and \(y\) the price of a burger.
So we have the following system of equations:
$$
\begin{cases}
5x + 3y = 29.99 \\
3x + 7y = 32.71
\end{cases}
$$
Can you find \(x\) and \(y\)?

Answer 2

ummm im not sure how to find x and y :( can you help...

Answer 3

ok, you can for example take the first equation
$$ 5x + 3y = 29.99 $$
Now let's find 'x' in that:
$$
5x = 29.99 - 3y\\
x = \frac{29.99 - 3y}{5}
$$
Now you can take value for 'x' and substitute with the 'x' in the second equation.
Then you'll have an equation with single variable 'y' =]
Once you find the numerical value of 'y' you can substitute it in the first equation and find such numerical value for 'x'.
Clear enough?

Answer 4

I don't know how to solve the equation you gave me though..

Answer 5

x=29.99−3y
5

this equation..

Answer 6

that's ok. the value of 'x', by the first equation, has to be equal to:
$$ x = \frac{29.99 -3y}{5} $$
We can't get it any better than that. But, we can take this value and put it in the second equation:
$$
3x + 7y = 32.71 \\
3\cdot\bigg( \frac{29.99 -3y}{5} \bigg) + 7y = 32.71
$$
Can you solve this one for 'y'?

Answer 7

so.. if I solve the distributive form.. i get 89.97 - 9y over 15 right?

Answer 8

and.. then..i divide 89.97 with 15? and get 5.998..

Answer 9

eh, why divide by 15? we get (89.97 - 9y) / 5 for first term

Answer 10

oh..bhhaha ok..:/

Answer 11

and..do I have to add the two ys? omgsh. i have no idea.

Answer 12

let's now multiply this by 5 to get rid of the factorial
we should now now
$$
\cancel 5 \cdot \bigg(\frac{89.97 - 9y}{\cancel5}\bigg) + 5 \cdot 7y = 5 \cdot 32.71
$$

Answer 13

Can you find 'y' now?

Answer 14

yea i think so :D wait,,,,

Answer 15

um first i multiplied 5 times 32. 71 and I got 163.55

Answer 16

then I have to add the two numbers without a y next to it right..?

Answer 17

ok so far =)

Answer 18

well ye, move and switch the sign. (basically subtracting 89.97 from both sides..)

Answer 19

oh so i subtract 89.97 with 163.55 which is 735.8

Answer 20

73.58 ye

Answer 21

oh oops yeah. and do i move the five over too?

Answer 22

no. we have 5*7y. which means 35*y, right?

Answer 23

we need to add to that the remaining (-9*y) from the factorial we had.

Answer 24

to clarify what we have now is basically:
$$
89.97 - 9y + 5\cdot7y = 163.55 \\
- 9y + 5\cdot7y = 163.55 - 89.97 \\
-9y + 35y = 73.58
$$

Answer 25

-9y + 35y becomes what?

Answer 26

26y

Answer 27

so the answer is 2.83

Answer 28

yes. now let's take this and put the new value of 'y' into the first equation
$$
5x + 3y = 29.99
$$

Answer 29

so.. x is 4.3?

Answer 30

yes
here, same solution as wolfram alpha =]
http://www.wolframalpha.com/input/?i=5x%2B3y+%3D+29.99%2C+3x%2B7y+%3D+32.71

Answer 31

thank you so much :D

Answer 32

well, half way. after I read that your question asks to solve it... I've seen that they also want you to use Cramer's Rule doing so..

Answer 33

oh... ok

Answer 34

$5.30
$1.83
$4.30
$2.83
there are four answers though

Answer 35

so wouldn't it be the third one?

Answer 36

Solving as we did is the 'simple' way for solving system of equations. you should be familiar with it first.
If you only need the solution then you have them already. x was packet of cheese and y was burger.
If you also need the way... then we can use Cramer's Rule.

Answer 37

I think I like the simple way better. thanks though :D

Answer 38

To be honest, I like it better too heh =)
well you're welcome, good luck with the rest

Answer 39

thanksss :DD