SOLUTION: Ken bought three chocolate bars and a pack of gum and paid $1.75. Sally bought two chocolate bars and four packs of gum and paid $2.00. Find the cost of a chocolate bar and the c

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: Ken bought three chocolate bars and a pack of gum and paid $1.75. Sally bought two chocolate bars and four packs of gum and paid $2.00. Find the cost of a chocolate bar and the c      Log On


   

Question 109228: Ken bought three chocolate bars and a pack of gum and paid $1.75. Sally bought two chocolate bars and four packs of gum and paid $2.00. Find the cost of a chocolate bar and the cost of a pack of gum.

3x +1y =$1.75
2x + 4y = $2.00
I know I have to multiply the second equation by a number in order to eliminate the variable x when added but I can't figure what number to use that will equal 3x. Please help
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Note: I'm going to write 1.75 in fraction form as 7%2F4

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


%283%29%2Ax%2B%281%29%2Ay=7%2F4 Start with the first equation


4%28%283%29%2Ax%2B%281%29%2Ay%29=%284%29%2A%287%2F4%29 Multiply both sides by the LCD 4



12%2Ax%2B4%2Ay=7Distribute and simplify


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Lets start with the given system of linear equations

12%2Ax%2B4%2Ay=7
2%2Ax%2B4%2Ay=2

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 12 and 2 to some equal number, we could try to get them to the LCM.

Since the LCM of 12 and 2 is 12, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -6 like this:

1%2A%2812%2Ax%2B4%2Ay%29=%287%29%2A1 Multiply the top equation (both sides) by 1
-6%2A%282%2Ax%2B4%2Ay%29=%282%29%2A-6 Multiply the bottom equation (both sides) by -6


So after multiplying we get this:
12%2Ax%2B4%2Ay=7
-12%2Ax-24%2Ay=-12

Notice how 12 and -12 add to zero (ie 12%2B-12=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2812%2Ax-12%2Ax%29%2B%284%2Ay-24%2Ay%29=7-12

%2812-12%29%2Ax%2B%284-24%29y=7-12

cross%2812%2B-12%29%2Ax%2B%284-24%29%2Ay=7-12 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-20%2Ay=-5

y=-5%2F-20 Divide both sides by -20 to solve for y



y=1%2F4 Reduce


Now plug this answer into the top equation 12%2Ax%2B4%2Ay=7 to solve for x

12%2Ax%2B4%281%2F4%29=7 Plug in y=1%2F4


12%2Ax%2B4%2F4=7 Multiply



12%2Ax%2B1=7 Reduce



12%2Ax=7-1 Subtract 1 from both sides

12%2Ax=6 Combine the terms on the right side

cross%28%281%2F12%29%2812%29%29%2Ax=%286%29%281%2F12%29 Multiply both sides by 1%2F12. This will cancel out 12 on the left side.


x=1%2F2 Multiply the terms on the right side


So our answer is

x=1%2F2, y=1%2F4

which also looks like

(1%2F2, 1%2F4)

Notice if we graph the equations (if you need help with graphing, check out this solver)

12%2Ax%2B4%2Ay=7
2%2Ax%2B4%2Ay=2

we get



graph of 12%2Ax%2B4%2Ay=7 (red) 2%2Ax%2B4%2Ay=2 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (1%2F2,1%2F4). This verifies our answer.


Since our answer is x=1%2F2 and y=1%2F4, these values in decimal form are

x=0.5 and y=0.25


So the chocolate bars are 50 cents and the gum is 25 cents