SOLUTION: 1) Solve the system of linear equations {{{system( x-2y+3z=-1, 4x+5y+z=26, -x+y-2z=-1)}}} a) What is the value of z? b) What is the value of x? c) What is the value of y?

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: 1) Solve the system of linear equations {{{system( x-2y+3z=-1, 4x+5y+z=26, -x+y-2z=-1)}}} a) What is the value of z? b) What is the value of x? c) What is the value of y?      Log On


   

Question 217226: 1) Solve the system of linear equations
system%28+x-2y%2B3z=-1%2C+4x%2B5y%2Bz=26%2C+-x%2By-2z=-1%29
a) What is the value of z?
b) What is the value of x?
c) What is the value of y?
Answer by ichudov(507) About Me  (Show Source):
You can put this solution on YOUR website!
system%28+x-2y%2B3z=-1%2C+4x%2B5y%2Bz=26%2C+-x%2By-2z=-1%29
From the last equation -x+y-2z=-1, you quickly get x = y-2z+1.
substitute this in first two and simplify
first equation : (y-2z+1)-2y+3z=1 or -y+z=0
second equation: 4(y-2z+1)+5y+z=26, simplify this:
Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:


YOUR ANSWER


  • Graphical form: 4%2A%28y-2z%2B1%29%2B5y%2Bz=26 simplifies to 9%2Ay-7%2Az-22=0
  • Text form: 4*(y-2z+1)+5y+z=26 simplifies to 9*y-7*z-22=0
  • Cartoon (animation) form: simplify_cartoon%28+4%2A%28y-2z%2B1%29%2B5y%2Bz=26+%29
    For tutors: simplify_cartoon( 4*(y-2z+1)+5y+z=26 )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at 4%2A%28y-2%2Az%2B1%29%2B5%2Ay%2Bz=highlight_red%28+26+%29.
Moved these terms to the left highlight_green%28+-26+%29
It becomes 4%2A%28y-2%2Az%2B1%29%2B5%2Ay%2Bz-highlight_green%28+26+%29=0.
Look at highlight_red%28+4%2A%28y-2%2Az%2B1%29+%29%2B5%2Ay%2Bz-26=0.
Expanded term 4 by using associative property on %28y-2%2Az%2B1%29
It becomes .
Look at 4%2Ay-highlight_red%28+4+%29%2Ahighlight_red%28+2+%29%2Az%2B4%2A1%2B5%2Ay%2Bz-26=0.
Multiplied numerator integers
It becomes 4%2Ay-highlight_green%28+8+%29%2Az%2B4%2A1%2B5%2Ay%2Bz-26=0.
Look at 4%2Ay-8%2Az%2Bhighlight_red%28+4+%29%2Ahighlight_red%28+1+%29%2B5%2Ay%2Bz-26=0.
Multiplied numerator integers
It becomes 4%2Ay-8%2Az%2Bhighlight_green%28+4+%29%2B5%2Ay%2Bz-26=0.
Look at 4%2Ay-8%2Az%2Bhighlight_red%28+4+%29%2B5%2Ay%2Bz-highlight_red%28+26+%29=0.
Added fractions or integers together
It becomes 4%2Ay-8%2Az%2Bhighlight_green%28+-22+%29%2B5%2Ay%2Bz=0.
Look at 4%2Ay-8%2Az%2Bhighlight_red%28+-22+%29%2B5%2Ay%2Bz=0.
Moved -22 to the right of expression
It becomes 4%2Ay-8%2Az%2B5%2Ay%2Bz%2Bhighlight_green%28+-22+%29=0.
Look at 4%2Ay-8%2Az%2B5%2Ay%2Bz%2Bhighlight_red%28+-22+%29=0.
Removed extra sign in front of -22
It becomes 4%2Ay-8%2Az%2B5%2Ay%2Bz-highlight_green%28+22+%29=0.
Look at highlight_red%28+4%2Ay+%29-8%2Az%2Bhighlight_red%28+5%2Ay+%29%2Bz-22=0.
Eliminated similar terms highlight_red%28+4%2Ay+%29,highlight_red%28+5%2Ay+%29 replacing them with highlight_green%28+%284%2B5%29%2Ay+%29
It becomes highlight_green%28+%284%2B5%29%2Ay+%29-8%2Az%2Bz-22=0.
Look at %284%2B5%29%2Ay-highlight_red%28+8%2Az+%29%2Bhighlight_red%28+z+%29-22=0.
Eliminated similar terms highlight_red%28+-8%2Az+%29,highlight_red%28+z+%29 replacing them with highlight_green%28+%28-8%2B1%29%2Az+%29
It becomes %284%2B5%29%2Ay%2Bhighlight_green%28+%28-8%2B1%29%2Az+%29-22=0.
Look at %28highlight_red%28+4+%29%2Bhighlight_red%28+5+%29%29%2Ay%2B%28-8%2B1%29%2Az-22=0.
Added fractions or integers together
It becomes %28highlight_green%28+9+%29%29%2Ay%2B%28-8%2B1%29%2Az-22=0.
Look at %289%29%2Ay%2B%28-highlight_red%28+8+%29%2Bhighlight_red%28+1+%29%29%2Az-22=0.
Added fractions or integers together
It becomes %289%29%2Ay%2B%28highlight_green%28+-7+%29%29%2Az-22=0.
Look at %289%29%2Ay%2B%28highlight_red%28+-7+%29%29%2Az-22=0.
Removed extra sign in front of -7
It becomes %289%29%2Ay%2B%28-highlight_green%28+7+%29%29%2Az-22=0.
Look at highlight_red%28+%28highlight_red%28+9+%29%29%2Ay+%29%2B%28-7%29%2Az-22=0.
Remove unneeded parentheses around factor highlight_red%28+9+%29
It becomes highlight_green%28+9+%29%2Ay%2B%28-7%29%2Az-22=0.
Look at 9%2Ay%2Bhighlight_red%28+%28-highlight_red%28+7+%29%29%2Az+%29-22=0.
Remove unneeded parentheses around factor highlight_red%28+7+%29
It becomes 9%2Ay-highlight_green%28+7+%29%2Az-22=0.
Result: 9%2Ay-7%2Az-22=0

Universal Simplifier and Solver


Done!


so we get +system%28+-y%2Bz=0%2C+9y-7z=22%29+
Solve it
Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
+system%28+%0D%0A++++-1%5Cy+%2B+1%5Cz+=+0%2C%0D%0A++++9%5Cy+%2B+-7%5Cz+=+22+%29%0D%0A++We'll use substitution. After moving 1*z to the right, we get:
-1%2Ay+=+0+-+1%2Az, or y+=+0%2F-1+-+1%2Az%2F-1. Substitute that
into another equation:
9%2A%280%2F-1+-+1%2Az%2F-1%29+%2B+-7%5Cz+=+22 and simplify: So, we know that z=11. Since y+=+0%2F-1+-+1%2Az%2F-1, y=11.

Answer: system%28+y=11%2C+z=11+%29.


Since y and z = 11, and x = y-2z+1, we get x = 11-22+1 = -10