Solve the following system of equations:
4x+y–2z=10
2x–y+2z=–9
x+y+z=6
I noticed that the coefficients of y are ± 1 , so ...
add the first and the second:
6x = 1
x = 1/6 , well that was lucky
add the second and third:
3x + 3z = -3
x + z = -1
1/6 + z = -1
z = -7/6
sub into the last:
1/6 + y - 7/6 = 6
times 6
1 + 6y - 7 = 36
y = 42/6 = 7
4x+y–2z=10
2x–y+2z=–9
-----------add
6 x = 1
x = 1/6
1/6 + y + z = 6
is
6 y + 6 z = 35
2(1/6)-y+2z = -9 --> -3 y + 6 z = -28
so use those two
6 y + 6 z = 35
-3y + 6 z = -28
--------------------subtract
9 y = 63
y = 7
1/6 + 7 + z = 6
1 + 42 + 6 z = 36
6 z = 36-43 = -7
z - -7/6
To solve the system of equations, we can use the method of elimination or substitution. I will explain how to solve them using elimination.
Step 1: Multiply the second equation by a suitable constant to make the coefficients of z the same but with opposite signs.
- In this case, multiplying the second equation by 2 will make the coefficients of z the same (-2z and 2z) but with opposite signs:
2(2x – y + 2z) = 2(-9)
4x - 2y + 4z = -18
Step 2: Add the two equations together to eliminate z.
(4x + y - 2z) + (4x - 2y + 4z) = 10 + (-18)
8x - y = -8
Step 3: Multiply the first equation by a suitable constant to make the coefficients of z the same but with opposite signs.
- In this case, multiplying the first equation by 2 will make the coefficients of z the same (-2z and 2z) but with opposite signs:
2(4x + y - 2z) = 2(10)
8x + 2y - 4z = 20
Step 4: Add the new equation obtained in Step 3 to the third equation to eliminate z.
(8x + 2y - 4z) + (x + y + z) = 20 + 6
9x + 3y = 26
Step 5: Simplify and solve the system of equations formed in Step 2 and Step 4:
8x - y = -8 ...(1)
9x + 3y = 26 ...(2)
To eliminate y, we can multiply equation (1) by 3 and equation (2) by 1, then subtract:
3(8x - y) = 3(-8)
9x - 3y = -24
(9x + 3y) - (9x - 3y) = 26 - (-24)
6y = 50
y = 50/6
y = 25/3
Step 6: Substitute the value of y back into equation (1) to solve for x:
8x - (25/3) = -8
8x = -8 + (25/3)
8x = -24/3 + (25/3)
8x = 1/3
x = 1/24
Step 7: Substitute the values of x and y back into one of the original equations, such as the third equation, to solve for z:
(1/24) + (25/3) + z = 6
z = 6 - (1/24) - (25/3)
z = 143/24
Therefore, the solution to the system of equations is:
x = 1/24, y = 25/3, z = 143/24.