SOLUTION: Solve the following system of equations for the unknown variables. 2x + y = 17 y + z = 6 x – z = 7

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Solve the following system of equations for the unknown variables. 2x + y = 17 y + z = 6 x – z = 7       Log On


   

Question 143829: Solve the following system of equations for the unknown variables.
2x + y = 17
y + z = 6
x – z = 7

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x-z+=+7 Start with the third equation


z=x-7 Solve for z


y+%2B+z+=+6+ Go to the 2nd equation


y+%2B+x-7+=+6+ Plug in z=x-7


y+%2B+x=+13 Add 7 to both sides


x%2By=+13 Rearrange the terms


So we now have the system

2x+%2B+y+=+17
x%2By=+13



Let's use substitution to solve this system


Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

2x%2By=17 Start with the first equation


y=17-2x Subtract 2x from both sides


y=-2x%2B17 Rearrange the equation


y=%28-2x%2B17%29%2F%281%29 Divide both sides by 1


y=%28%28-2%29%2F%281%29%29x%2B%2817%29%2F%281%29 Break up the fraction


y=-2x%2B17 Reduce



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Since y=-2x%2B17, we can now replace each y in the second equation with -2x%2B17 to solve for x



x%2Bhighlight%28%28-2x%2B17%29%29=13 Plug in y=-2x%2B17 into the first equation. In other words, replace each y with -2x%2B17. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



-x%2B17=13 Combine like terms on the left side


-x=13-17Subtract 17 from both sides


-x=-4 Combine like terms on the right side


x=%28-4%29%2F%28-1%29 Divide both sides by -1 to isolate x



x=4 Divide





So the first part of our answer is: x=4









Since we know that x=4 we can plug it into the equation y=-2x%2B17 (remember we previously solved for y in the first equation).



y=-2x%2B17 Start with the equation where y was previously isolated.


y=-2%284%29%2B17 Plug in x=4


y=-8%2B17 Multiply


y=9 Combine like terms




So the second part of our answer is: y=9


x-z+=+7 Go back to the third equation


4-z+=+7 Plug in x=4




-z=7-4Subtract 4 from both sides


-z=3 Combine like terms on the right side


z=%283%29%2F%28-1%29 Divide both sides by -1 to isolate z



z=-3 Divide






So the third part of our answer is: z=-3



-----------------Summary------------------------------

So our answers are:

x=4, y=9, and z=-3