$x + y + z = - 1, 3 x + y + 4 z = 8, - x - y + 7 z = 9$ $x+y+z=-1, 3x+y+4z=8,-x-y+7z=9$ ?

Question

$x + y + z = - 1, 3 x + y + 4 z = 8, - x - y + 7 z = 9$ $x+y+z=-1, 3x+y+4z=8,-x-y+7z=9$ ?

2 Answers

Impact of this question

3899 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-09T18:19:41

$x = 3$ $x=3$
$y = - 5$ $y=-5$
$z = 1$ $z=1$

Explanation:

There are three equations with three variables.

Make $y$ $y$ the subject in all three equations:

$y = - x - z - 1$ $y= -x-z -1" "$ ..... equation 1
$y = - 3 x - 4 z + 8$ $y = -3x-4z+8" "$ ... equation 2
$y = - x + 7 z - 9$ $y= -x+7z-9" "$ ...equation 3

By equating the equations in pairs we can form two equations with the variables $x and z$ $x and z$ and solve them simultaneously

Using equations 1 and 2: $y = y$ $" "y=y$

$- x - z - 1 = - 3 x - 4 z + 8$ $" "-x-z-1 = -3x-4z+8$
$3 x - x + 4 z - z = 8 + 1 \leftarrow$ $3x-x+4z-z=8+1" "larr$ re-arrange

$2 x + 3 z = 9$ $2x+3z=9" "$ equation A

Using equations 3 and 2 $y = y$ $" "y=y$

$- x + 7 z - 9 = - 3 x - 4 z + 8 \leftarrow$ $" "-x+7z-9=-3x-4z+8" "larr$ re-arrange

$3 x - x + 7 z + 4 z = 8 + 9$ $3x-x+7z+4z=8+9$

$2 x + 11 z = 17$ $2x+11z=17" "$ equation B

Now solve A and B for $x and z$ $x and z$

$2 x + 11 z = 17 m m m m m m m m m m m A$ $" "2x+11z = 17color(white)(mmmmmmmmmmm)A$
$2 x + 3 z = 9 m m m m m m m m m m m m B$ $" "2x+3z =9color(white)(mmmmmmmmmmmm)B$

$A - B : 8 z = 8$ $A-B:" "8z = 8$
$m m m m m m z = 1$ $color(white)(mmmmmm)z =1$

$2 x + 3 (1) = 9$ $2x +3(1)=9$
$2 x + 3 = 9$ $2x +3=9$
$2 x = 6$ $2x =6$
$x = 3$ $x=3$

Now find $y$ $y$ from equation 1
$y = - x - z - 1$ $y= -x-z -1$
$y = - (3) - (1) - 1$ $y = -(3)-(1)-1$
$y = - 5$ $y =-5$

Check with equation 2

$y = - 3 x - 4 z + 8$ $y =-3x-4z+8$
$y = - 3 (3) - 4 (1) + 8$ $y = -3(3)-4(1)+8$
$y = - 9 - 4 + 8$ $y=-9-4+8$
$y = - 5$ $y=-5$

Answer 2 · 2017-11-09T19:51:46

$x = 3$ $x=3$ , $y = - 5$ $y=-5$ and $z = 1$ $z=1$

Explanation:

$x + y + z = - 1$ $x+y+z=-1$ , $3 x + y + 4 z = 8$ $3x+y+4z=8$ an $- x - y + 7 z = 9$ $-x-y+7z=9$

From first equation, $z = - x - y - 1$ $z=-x-y-1$

Plug $z$ $z$ into second an third ones;

$3 x + y + 4 \cdot (- x - y - 1) = 8$ $3x+y+4*(-x-y-1)=8$

$3 x + y - 4 x - 4 y - 4 = 8$ $3x+y-4x-4y-4=8$

$- x - 3 y = 12$ $-x-3y=12$

$- x - y + 7 \cdot (- x - y - 1) = 9$ $-x-y+7*(-x-y-1)=9$

$- x - y - 7 x - 7 y - 7 = 9$ $-x-y-7x-7y-7=9$

$- 8 x - 8 y = 16$ $-8x-8y=16$

$- 8 \cdot (x + y) = 16$ $-8*(x+y)=16$ or $x + y = - 2$ $x+y=-2$

From second one, $x = - 3 y - 12$ $x=-3y-12$

Plug $x$ $x$ into third one;

$(- 3 y - 12) + y = - 2$ $(-3y-12)+y=-2$

$- 2 y - 12 = - 2$ $-2y-12=-2$

$- 2 y = 10$ $-2y=10$ , so $y = - 5$ $y=-5$

Hence $x = - 3 y - 12 = (- 3) \cdot (- 5) - 12 = 3$ $x=-3y-12=(-3)*(-5)-12=3$

Thus, $z = - x - y - 1 = - 3 - (- 5) - 1 = 1$ $z=-x-y-1=-3-(-5)-1=1$

Science

Math

Humanities

... and beyond

$x + y + z = - 1, 3 x + y + 4 z = 8, - x - y + 7 z = 9$ $x+y+z=-1, 3x+y+4z=8,-x-y+7z=9$ ?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

x+y+z=−1,3x+y+4z=8,−x−y+7z=9x+y+z=-1, 3x+y+4z=8,-x-y+7z=9?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

$x + y + z = - 1, 3 x + y + 4 z = 8, - x - y + 7 z = 9$ $x+y+z=-1, 3x+y+4z=8,-x-y+7z=9$ ?