1. Solve any equation for any letter.
2. Substitute what that letter equals for that letter in the
other two equations and simplify.
[Now we'll have only a system in 2 equations and 2 letters]
3. Solve either equation for either letter.
4. Substitute what that letter equals for that letter in the
other equation in 2 letters and simplify.
5. Substitute that in the other letter in one of the
other equations in two letters and solve for a
second letter.
6. Finally substitute the numbers for the two letters
you have in one of the original equations and solve
for the last remaining letter:
(eq. 1) 2x - y - z = 15
(eq. 2) 4x + 5y + 2z = 10
(3q. 3) -x - 4y + 3z = -20
1. Solve any equation for any letter.
I'll pick an easiest equation to solve for an easy letter.
Hmmm. Think I'll pick eq. 1 to solve for, hmmm, y
(eq. 1) 2x - y - z = 15
-y = 15 - 2x + z
y = -15 + 2x - z
2. Substitute what that letter equals for that letter in the
other two equations and simplify:
(eq. 2) 4x + 5y + 2z = 10
4x + 5(-15 + 2x - z) + 2z = 10
4x - 75 + 10x - 5z + 2z = 10
14x - 3z - 75 = 10
(eq. 4) 14x - 3z = 85
(3q. 3) -x - 4y + 3z = -20
-x - 4(-15 + 2x - z) +3z = -20
-x + 60 - 8x + 4z + 3z = -20
-9x + 7z + 60 = -20
(eq. 5) -9x + 7z = -80
Now we have only a system in 2 equations and 2 unknowns:
(eq. 4) 14x - 3z = 85
(eq. 5) -9x + 7z = -80
3. Solve either equation for either letter.
I'll pick the easier equation to solve for the easier letter.
(eq. 4) 14x - 3z = 85
-3z = 85 - 14x
z = -85/3 + (14/3)x
4. Substitute what that letter equals for that letter in the
other equation and simplify:
(eq. 5) -9x + 7z = -80
-9x + 7[-85/3 + (14/3)x] = -80
-9x - 595/3 + (98/3)x = -80
-27x - 595 + 98x = -240
71x - 595 = -240
71x = 355
x = 5
5. Substitute that in the other letter in one of the
other equations in two letters and solve for a
second letter:
I'll substitute x = 5 in eq. 5:
(eq. 5) -9x + 7z = -80
-9(5) + 7z = -80
-45 + 7z = -80
7z = =35
z = -5
6. Finally substitute the numbers for the two letters
you have in one of the original equations and solve
for the last remaining letter:
I'll pick eq. 3 to substitute x = 5 and z = -5 in
(3q. 3) -x - 4y + 3z = -20
-(5) - 4y + 3(-5) = -20
-5 - 4y - 15 = -20
-4y - 20 = -20
-4y = 0
y = 0
Solution: (x,y,z) = (5,0,-5)
Follow those same steps in your second problem.
Answers are (x,y,z) = (-6,-20,8)
However your third problem is different.
x + 3y - z = 12
2x + 4y - 2z = 6
-x - 2y + z = -6
1. Solve any equation for any letter.
I'll pick the first to solve for x
x = 12 - 3y + z
2. Substitute what that letter equals for that letter in the
other two equations and simplify.
2(12 - 3y + z) + 4y - 2z = 6
24 - 6y + 2z + 4y - 2z = 6
24 -2y = 6
-2y = -18
y = 9
-x - 2y + z = -6
-(12 - 3y + z) - 2y + z = -6
-12 + 3y - z - 2y + z = -6
-12 + y = -6
y = 6
y cannot equal both 6 and 9.
So the system is inconsistent. It has no solution.
Edwin
