Which of the following system of equations are easier to solve:
System of equations 1:
x + 3y + z = 5
-2x + 2y = 3
3x + 3y + 2z = 9
System of equations 2:
x – 3y + 2z = 6
y + 3z = 12
z = 2
System of equations 2 is easier to solve. This system of equations is in row-echelon form which means it has a “stair-step” with leading coefficients of 1. To solve such a system you use a method called back substitution.
Solving with back substitution:
Step 1: Plug z = 2 into y + 3z = 12 to solve for y.
y + 3(2) = 12
y + 6 = 12
y = 6
Step 2: Plug z = 2 and y = 6 into x – 3y + 2z = 6 to solve for x.
x – 3(6) + 2(2) = 6
x – 18 + 4 = 6
x – 22 = 6
x = 28
The solution is x=28, y=6, and z=2
To solve a system of equations not in row-echelon form, you must first convert it to row-echelon form. Using elimination to rewrite the system of equation 1 into row-echelon form.
x + 3y + z = 5 equation 1
-2x + 2y = 3 equation 2
3x + 3y + 2z = 9 equation 3
Step 1: Eliminate the x from equation 2 and equation 3.
To eliminate the x from -2x + 2y = 3, multiply the first equation by 2 and add it to the equation 2.
2(x + 3y + z = 5) –> 2x + 6y + 2z = 10
2x + 6y + 2z = 10
+ -2x + 2y = 3
8y + 2z = 13
To eliminate the x from 3x + 3y + 2z = 9, multiply the first equation by -3 and add it to the equation 2.
-3(x + 3y + z = 5) –> -3x – 9y – 3z = -15
-3x – 9y – 3z = -15
+ 3x + 3y + 2z = 9
6y – z = -6
New set of equations:
x + 3y + z = 5 equation 1
8y + 2z = 13 equation 2
6y – z = -6 equation 3
Step 2: Eliminate the y from equation 3. To do this multiply equation 2 by -3/4 and add it to equation 3 so you can cancel the y from equation 3.
-3/4 (8y + 2z = 13) –> -6y – 1.5z = -9.75
6y – z = -6
+ -6y – 1.5z = -9.75
– 2.5z = -15.75
solve for z
z = -13.25
New set of equations:
x + 3y + z = 5 equation 1
8y + 2z = 13 equation 2
z = -13.25 equation 3
Step 3: make sure all leading coefficients are 1. In this case 8y needs to be converted to just y. To do this simply multiply that equation by 1/8.
1/8 (8y + 2z = 13) = y + 1/4z = 1.625
New set of equations:
x + 3y + z = 5 equation 1
y + 1/4z = 1.625 equation 2
z = -13.25 equation 3
Step 4: Use back substitution to solve.
z = -13.25
y + 1/4z = 1.625
y + 1/4 (-13.25) = 1.625
y + 3.3125 = 1.625
y = -1.6875
x + 3y + z = 5
x + 3(-1.6875) + (-13.25) = 5
x – 18.3125 = 5
x = 23.3125
The solution is x = 23.3125, y = -1.6875, and z = -13.25