Can you manipulate the coefficients of any variable terms such that they could be eliminated?
(2,-1,1)
Practice makes perfect
The given system consists of equations of planes.
Notice that the coefficient of z in the first equation is the additive inverse of the coefficient of z in the third equation; they will add to be 0. Let's use the Elimination Method to find a solution to this system.
x+4y- 5z=-7 & (I) 3x+2y+3z=7 & (II) 2x+y+ 5z=8 & (III)
We can start by adding the third equation to the first equation to eliminate the z-terms.
x+4y-5z=-7 & (I) 3x+2y+3z=7 & (II) 2x+y+5z=8 & (III)
Having eliminated the z-variable from the first equation, we can continue by creating additive inverse coefficients for z in the second and third equations. Then we can add or subtract these equations to eliminate z from the second equation.
Next, we use our two equations that are only in terms of x and y to solve for the value of one of the variables. We will once again apply the Elimination Method, but this time will be similar to when using it in a system with only two variables.
The solution to the system is ( 2, -1, 1). This is the singular point at which all three planes intersect.
Now we can check our solution by substituting the values into the system.
x+4y-5z=-7 & (I) 3x+2y+3z=7 & (II) 2x+y+5z=8 & (III)
