2x+y+2z=25
3x-y+6z=48
5x+2y+4z=55
I got stuck after step 2 of trying to eliminate the first two equations where do I go from their and would kindly like full explanation of how to solve it so I can practice it out. Thank you :)
If you want to see the details of elimination, a good place to go is
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
A preliminary glance at the coefficients tells you that the y's would be the easiest to eliminate.
As a matter of fact, you can do it in your head:
add the first two equations:
5x + 8z = 73 , #4
mentally double the 2nd and add to the third:
11x + 16z = 151 , #5
double #4 ---> 10x + 16z = 146
subtract from #5
------> x = 5
back in #4
25+8z = 73
8z =48
z = 6
back in #1
2x+y+2z = 25
10 + y + 12 = 25
y = 3
To solve the system of equations, you can use the method of elimination or substitution. I'll explain the elimination method step by step:
Step 1: Write down the given system of equations:
2x + y + 2z = 25 ......(Equation 1)
3x - y + 6z = 48 ......(Equation 2)
5x + 2y + 4z = 55 ......(Equation 3)
Step 2: Choose two equations and eliminate one variable at a time. Let's eliminate the variable 'y' in equations 1 and 2.
Multiply Equation 1 by 1 (to maintain the equation):
2x + y + 2z = 25
Multiply Equation 2 by 1 (to maintain the equation):
3x - y + 6z = 48
Step 3: Add the modified Equation 1 and Equation 2 to eliminate 'y':
(2x + y + 2z) + (3x - y + 6z) = 25 + 48
2x + 3x + y - y + 2z + 6z = 73
5x + 8z = 73 ......(Equation 4)
Step 4: Let's eliminate the variable 'y' in equations 2 and 3.
Multiply Equation 2 by 2 (to maintain the equation):
3x - y + 6z = 48
Multiply Equation 3 by 1 (to maintain the equation):
5x + 2y + 4z = 55
Step 5: Add the modified Equation 2 and Equation 3 to eliminate 'y':
(3x - y + 6z) + (5x + 2y + 4z) = 48 + 55
3x + 5x - y + 2y + 6z + 4z = 103
8x + z = 103 ......(Equation 5)
So, after eliminating 'y', we obtained two new equations:
5x + 8z = 73 ......(Equation 4)
8x + z = 103 ......(Equation 5)
Now, you can solve these two equations simultaneously using any method, such as substitution or elimination, to find the values of 'x' and 'z'. Once you have the values of 'x' and 'z', you can substitute them back into any of the original equations, like Equation 1, to find the value of 'y'.
I hope this helps! Let me know if you have any further questions.