system of 3 linear equations (check my work please)

[Chase2001]

Solve the following system of linear equations.

 

a) [latex]2x-y+z=1[/latex]

b) [latex]4x+y+z=2[/latex]

c) [latex]x-y-2z=0[/latex]

 

The y's cancel in equations a and b to give e) [latex]6x+2z=3[/latex]

The y's cancel in equations b and c to give f) [latex]5x-z=3[/latex]

 

I now multiply f by 2 and add it to equation e to eliminate z giving me [latex]16x=7 \Rightarrow x=\frac{7}{16}[/latex]

 

Now I plug [latex]\frac{7}{16}[/latex] back into equation f giving me [latex]5\left(\frac{7}{16}\right)-z=2 \Rightarrow \frac{35}{16}-2=z \Rightarrow z=\frac{3}{16}[/latex]

 

Then finally I plug x and z back into equation b giving me [latex]4\left(\frac{7}{16}\right)+\frac{3}{16}+z=2 \Rightarrow \frac{35}{16}+\frac{3}{16}+z=2 \Rightarrow \frac{31}{16}-2=-z \Rightarrow z=\frac{1}{16}[/latex]

 

I find it difficult to keep track of these long winded questions so I took the time to learn latex so it should be easier for you guys to read and in turn easier to help me if need be.

 

Just out of curiosity what are linear equations used for in the real world? I would quite like to be a physicist or a structural engineer when I grow up

 

[studiot]

 

The y's cancel in equations b and c to give f)

 

 

Check this

 

You should always be able to check your own answers by back substituting your answers into the original equations.

[timo]

Didn't check the math (again ), but when plugging into equation b you probably want to plug in z for z, not for y. If you want to become a physicist (and perhaps also if you want to become an engineer) then learning tex is certainly recommended: pretty much every physicist knows and uses tex, so you can as well pick up that skill early.

 

I'd say that solving systems of linear equations is frequently done in physics, even though I cannot pinpoint a sensible example at the moment. There's two straightforward generalizations of it that are an asset in every physicist's math toolkit (which is why I tend not to realize the special case of systems of linear equations). The first one is linear algebra, which deals with structures called "matrices" that are similar to systems of linear equations. The other one is quite generic: Often, you have a system involving several unknown variables you want to know. What you often do is that you look for a sufficient number of known relations between the variables to be able to solve for their values. That is very similar to the concept of sets of linear equations, but generally the relations do not have to be linear.

[Endy0816]

Note, the written (f) above is off though the equation used later is correct. Likewise the 'z' written at the end should be labeled as y.

 

Plugging your numbers back into the equations results in:

 

(14 - 1 + 3)/16 = 16/16 = 1(Correct)

 

(28 + 1 + 3)/16 = 32/16 = 2(Correct)

 

(7 - 1 - 2(3))/16 = 0/16 = 0(Correct)

 

Your answer is correct, the labeling issue I suspect occurred during transposition. All in all a nice bit of work.

Edited February 4, 2014 by Endy0816

[Chase2001]

Note, the written (f) above is off though the equation used later is correct. Likewise the 'z' written at the end should be labeled as y.

 

Plugging your numbers back into the equations results in:

 

(14 - 1 + 3)/16 = 16/16 = 1(Correct)

 

(28 + 1 + 3)/16 = 32/16 = 2(Correct)

 

(7 - 1 - 2(3))/16 = 0/16 = 0(Correct)

 

Your answer is correct, the labeling issue I suspect occurred during transposition. All in all a nice bit of work.

 

Thanks