Solve the simultaneous questions:
(a) x+y+z=-3 (1)
x-y-2z=13 (2)
y-4z=25 (3)
(b)x+2y+3z=-14 (1)
2x+3y-z=13 (2)
3x-y+2z=-7 (3)
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
for the first one for example there are three rows and four columns
1 +1 +1 -3
1 -1 -2 13
1 +0 -4 25
x = 15/7
y = 4/7
z = -40/7
There is no unique or best way to solve these.
After doing a few you get a "feel" of what to look for.
e.g. for your first one, notice that last one says:
y - 4z = 25 , so y = 25+4z
- could be useful
look at the first two, they both have 1x
so lets subtract the 2nd from the 1st:
2y + 3z = - 16
remember y = 25+4z
2(25+4z) + 3z = -16
50 + 8z + 3z = -16
11z = -66
z = -6
then y = 25 + 4(-6) = 1
back in the 1st:
x+y+z=-3
x+1-6=-3
x = 2
x=2 , y=1 , z=-6
hint for the 2nd:
double the first and subtract from the 2nd to get a
y , z equation
triple the first and subtract from the 3rd to get another y , z equation
now solve as 2 equations in two unknowns
To solve a system of simultaneous equations like the ones given, we can use a method called Gaussian elimination. This involves using the equations to eliminate variables one by one.
(a) Let's start solving the first set of equations:
Step 1: Rearrange the equations so that the variables are aligned vertically.
(1) x + y + z = -3
(2) x - y - 2z = 13
(3) 0x + y - 4z = 25 (We can write it as 0x to align the variables)
Step 2: Choose one equation and perform operations to eliminate one variable from the other two equations. Let's choose equation (2) to eliminate the variable x.
Multiply equation (1) by -1: (-1)(x + y + z) = -1(-3) gives -x - y - z = 3
Subtract equation (2) from the new equation: (-x - y - z) - (x - y - 2z) = 3 - 13 simplifies to -3z = -10
Step 3: Solve the resulting equation for the variable that is left, in this case, z.
-3z = -10
Divide both sides by -3: z = 10/3 or approximately 3.33
Step 4: Substitute the value of z into one of the original equations to find the value of y. Let's use equation (3):
0x + y - 4(10/3) = 25 simplifies to y - (40/3) = 25
Add (40/3) to both sides: y = 25 + (40/3) = (75/3) + (40/3) = 115/3 or approximately 38.33
Step 5: Substitute the values of y and z back into one of the original equations to find the value of x. Let's use equation (1):
x + (115/3) + (10/3) = -3
Multiply through by 3 to eliminate fractions: 3x + 115 + 10 = -9
Combine like terms: 3x + 125 = -9
Subtract 125 from both sides: 3x = -134
Divide both sides by 3: x = -134/3 or approximately -44.67
So the solution to the first set of simultaneous equations is x = -44.67, y = 38.33, and z = 3.33.
(b) Let's now solve the second set of equations using the same method:
Step 1: Rearrange the equations so that the variables are aligned vertically.
(1) x + 2y + 3z = -14
(2) 2x + 3y - z = 13
(3) 3x - y + 2z = -7
Step 2: Choose one equation and perform operations to eliminate one variable from the other two equations. Let's choose equation (3) to eliminate the variable x.
Multiply equation (1) by -3: (-3)(x + 2y + 3z) = -3(-14) gives -3x - 6y - 9z = 42
Multiply equation (2) by 3: (3)(2x + 3y - z) = 3(13) gives 6x + 9y - 3z = 39
Add the new equations: (-3x - 6y - 9z) + (6x + 9y - 3z) = 42 + 39 simplifies to 3x + 3z = 81
Step 3: Solve the resulting equation for the variable that is left, in this case, z.
3x + 3z = 81
Subtract 3x from both sides: 3z = 81 - 3x
Step 4: Substitute the resulting expression for z into one of the original equations to find the value of y. Let's use equation (1):
x + 2y + 3(81 - 3x) = -14 simplifies to x + 2y + 243 - 9x = -14
Combine like terms: -8x + 2y = -257
Step 5: Continue to solve for the remaining variables using a method like substitution or elimination.