David W. answered • 06/08/15
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This word (story) problem asks for the weight of each type of box, Let’s assign variables:
L = weight of large box
S = weight of small box
The linear equations put coefficients with each variable and produce two functions:
3*L + 8*S = 97
5*L + 2*S = 88
Now, there are two methods to isolate one of the variables – (1) substitution, and (2) elimination.
Let’s use elimination. Multiplying the second equation by 4, we now have:
3*L + 8*S = 97
20*L + 8*S = 352
------------------------- (add or subtract equations.
L = weight of large box
S = weight of small box
The linear equations put coefficients with each variable and produce two functions:
3*L + 8*S = 97
5*L + 2*S = 88
Now, there are two methods to isolate one of the variables – (1) substitution, and (2) elimination.
Let’s use elimination. Multiplying the second equation by 4, we now have:
3*L + 8*S = 97
20*L + 8*S = 352
------------------------- (add or subtract equations.
To make it easy, subtract eq1 from eq2)
17*L + 0 = 255
L = 15 kg
Substitution this value of S into either equation to get L,
5*(15) + 2*S = 88
2*S = 13
S = 6.5 kg
Checking:
Is 3*(15) + 8*(6.5) = 97 ?
45 + 52 = 97 ? Yes
Is 5*(15) + 2*(6.5) = 88 ?
75 + 13 = 88 ? Yes
17*L + 0 = 255
L = 15 kg
Substitution this value of S into either equation to get L,
5*(15) + 2*S = 88
2*S = 13
S = 6.5 kg
Checking:
Is 3*(15) + 8*(6.5) = 97 ?
45 + 52 = 97 ? Yes
Is 5*(15) + 2*(6.5) = 88 ?
75 + 13 = 88 ? Yes
Andrew M.
the basic logic was simply that if we have 2 equations which are both true then we can add or subtract them and still have a true equation. For example:
2 + 7 = 9
3 + 10 = 13
Adding those together we get 5 + 17 = 22 which is also true
Or subtracting them we get -1 - 3 = -4 which is also true.
That is the rule David used for the idea of adding the two equations together to create one equation.
Also note that if a + b = c then ax + bx = cx
Meaning we can multiply every item in a true equation by the same factor and still have a true equation.
For example: 5 - 9 = -4
4(5) + 4(-9) = 4(-4)
20 - 36 = -16
That is what David did in order to solve this system.
First he multiplied the 2nd equation by 4 in order to have 8S in both equations. Then he subtracted
the 2nd equation from the 1st equation giving
-17L = - 355
L = -335/(-17)
L = 15 kg
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06/09/15
Andrew M.
Correction: When we subtracted the 2nd equation from the first we got
-17L = -255
L = (-255)/(-17)
L = 15 kg
Once we know the weight of the large box just put 15 place of L in either equation and solve for S
3(15) + 8S = 97
45 + 8S = 97
8S = 52
S = 52/8
S = 6.5 kg
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06/09/15
David W.
I often write using outline points like (a) this is important, and (b) this is pretty useful. In my answer, I wrote that (1) this is the first option, and (2) this is the second option.
When looking closely at the two equations, I wanted to find a number (and 4 was a good choice) that I could use to multiply one equation in order to get a result that would match the corresponding coefficient in the other equation and produce a zero when the equations were added or subtracted. This may be (a) unnecessary if the coefficients already match, (b) easy if you only have to multiply one equation, or (c) harder if you have to multiply both equations by different numbers -- much like finding a common denominator. [Hey! I just used my outline technique on purpose!] You should get the same result whichever choice you make, but you will learn to save time and make the solution easier when you see how your choice affects the rest of the calculations.
For practice, (1) use [a] and [b] in talking to someone [often they think I was a professor/teacher/tutor and (2) pick a number or numbers other than 4 to use to get common coefficients.
p.s., [I often write these too] THX for reading my answer so closely; that's a sign of a good student! [often, good students even point out my typos or other mistakes]. Best wishes!
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06/09/15
Fran L.
Thank you
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06/09/15
Fran L.
06/08/15