4x + 3y + 2z = 34; 2x + 4y + 3z = 45; 3x + 2y + 4z = 47 First, eliminate terms in z from 2 of the equations, by muliplying first equation by 2 and subtracting third equation from the answer: 8x + 6y + 4z = 68, subtract leaving 5x + 4y = 21 (equation 4) Similarly multiply the first equation by 3 and the second by 2 giving 12x + 9y + 6z = 102 and 4x + 8y + 6z = 90 Subtract again and we have 8x + y = 12 or y = 12 - 8x Substitute this in equation 4 gives 5x + 4(12 -8x) = 21 Simplify: 5x + 48 - 32x = 21 = -27x = -27 so x = 1 y = 12 - 8x so y = 4 and in one of the original equations 4 + 12 + 2z = 34, ie 2z = 34 -16 so z =9 Check: 2x + 4y + 3z = 2 +16 + 27 = 45 and 3x + 2y + 4z = 3 + 8 + 36 = 47 QED!
x = 4 and y = 0
x = -1.2, y = -3
To solve for two unknowns (x and y) it is necessary to have two independent equations.
Solve this system of equations. 5x+3y+z=-29 x-3y+2z=23 14x-2y+3z=-18 Write the solution as an ordered triple.
x = 6 and y = -2
To solve an equation with three unknowns, x, y and z, you require 3 independent equations.
When talking about a "system of equations", you would normally expect to have two or more equations. It is quite common to have as many equations as you have variables, so in this case you should have two equations.