what is the solution of the system of equations.
2x+2y+3z=-6
3x+5y+4z=3
2x+3y+4z=-10
(x,y,z) = (5,4,-8)
is there away to show your work
sure. One way is to do elimination by adding and subtracting multiples of one equation from another.
subtract 2*second from 3*first:
6x+6y+9z = -18
6x+10y+8z = 6
-----------------
-4y + z = -24
subtract first from third
-y - z = 4
now add those two to get
-5y = -20
y=4
so, z=-8 and x=5
To find the solution to the system of equations, we can use the method of solving systems of equations called Gaussian elimination.
Step 1: Write the augmented matrix of the system:
[2 2 3 | -6]
[3 5 4 | 3]
[2 3 4 | -10]
Step 2: Perform row operations to transform the augmented matrix into row-echelon form. The goal is to introduce zeros below the diagonal entries.
Multiply the first row by 3/2 and subtract it from the second row:
[2 2 3 | -6]
[0 1/2 -1/2 | 21/2]
[2 3 4 | -10]
Multiply the first row by 1 and subtract it from the third row:
[2 2 3 | -6]
[0 1/2 -1/2 | 21/2]
[0 1 1 | -2]
Step 3: Continue performing row operations to obtain a matrix in reduced row-echelon form. The goal is to have leading 1's in each row and zeros above and below these ones.
Multiply the second row by 2 and subtract it from the first row:
[2 0 4 | -15]
[0 1/2 -1/2 | 21/2]
[0 1 1 | -2]
Multiply the second row by 1 and subtract it from the third row:
[2 0 4 | -15]
[0 1/2 -1/2 | 21/2]
[0 0 3/2 | -25/2]
Step 4: Finally, express the row-echelon form as a system of equations:
2x + 4z = -15 ...(1)
0.5y - 0.5z = 10.5 ...(2)
1.5z = -12.5 ...(3)
From equation (3), we can solve for z:
z = -12.5 / 1.5
z = -25 / 3
Substituting the value of z into equation (2), we can solve for y:
0.5y - 0.5*(-25/3) = 10.5
0.5y + 12.5/3 = 10.5
0.5y = 10.5 - 12.5/3
0.5y = 10.5 - 4.1667
0.5y = 6.3333
y = 6.3333 / 0.5
y = 12.6667
Finally, substituting the values of y and z into equation (1), we can solve for x:
2x + 4*(-25/3) = -15
2x - 100/3 = -15
2x = -15 + 100/3
2x = -45/3 + 100/3
2x = 55/3
x = 55/3 / 2
x = 55/6
Therefore, the solution to the system of equations is:
x = 55/6
y = 12.6667
z = -25/3