x+y+z=11
499x+40y-95z= -46
x+2y-z= 0
solve for x, y, z
Answers
x + y + z = 11
499 x + 40 y - 95 z = -46
x + 2 y - z = 0
In the first equation, look to solve for z:
{x + y + z = 11
499 x + 40 y - 95 z = -46
x + 2 y - z = 0
Subtract y + x from both sides:
{z = (-y - x) + 11
499 x + 40 y - 95 z = -46
x + 2 y - z = 0
Substitute z = -y - x + 11 into the second and third equations:
{z = -y - x + 11
499 x - 95 (-y - x + 11) + 40 y = -46
3 y + 2 x - 11 = 0
499 x - 95 (-y - x + 11) + 40 y = 499 x + (95 y + 95 x - 1045) + 40 y = 135 y + 594 x - 1045:
{z = -y - x + 11
135 y + 594 x - 1045 = -46
3 y + 2 x - 11 = 0
In the second equation, look to solve for x:
{z = -y - x + 11
135 y + 594 x - 1045 = -46
3 y + 2 x - 11 = 0
Subtract 135 y - 1045 from both sides:
{z = -y - x + 11
594 x = 999 - 135 y
3 y + 2 x - 11 = 0
Divide both sides by 594:
{z = -y - x + 11
x = 37/22 - (5 y)/22
3 y + 2 x - 11 = 0
Substitute x = 37/22 - (5 y)/22 into the third equation:
{z = -y - x + 11
x = 37/22 - (5 y)/22
-11 + 2 (37/22 - (5 y)/22) + 3 y = 0
-11 + 2 (37/22 - (5 y)/22) + 3 y = 3 y + (37/11 - (5 y)/11) - 11 = (28 y)/11 - 84/11:
{z = -y - x + 11
x = 37/22 - (5 y)/22
(28 y)/11 - 84/11 = 0
In the third equation, look to solve for y:
{z = -y - x + 11
x = 37/22 - (5 y)/22
(28 y)/11 - 84/11 = 0
Bring (28 y)/11 - 84/11 together using the common denominator 11:
{z = -y - x + 11
x = 37/22 - (5 y)/22
(28 (y - 3))/11 = 0
Multiply both sides by 11/28:
{z = -y - x + 11
x = 37/22 - (5 y)/22
y - 3 = 0
Add 3 to both sides:
{z = -y - x + 11
x = 37/22 - (5 y)/22
y = 3
Substitute y = 3 into the first and second equations:
{z = 8 - x
x = 1
y = 3
Substitute x = 1 into the first equation:
{z = 7
x = 1
y = 3
Collect results in alphabetical order:
Answer: |
| {x = 1
y = 3
z = 7