The coefficient matrix is a matrix that contains only the coefficients of a system. Start by calculating |C|, the determinant of the coefficient matrix.
(2,-3,6)
Practice makes perfect
We will use Cramer's Rule to solve the given system of equations.
5x+ 2y+ 0z=4 3x+ 4y+ 2z=6 7x+ 3y+ 4z=29
Let C be the coefficient matrix of the system. The coefficient matrix is a matrix that contains only the coefficients of a system.
C=
[
ccc
5 & 2 & 0
3 & 4 & 2
7 & 3 & 4
]
If the determinant of the coefficient matrix, |C|, is different from zero, we can find the solution to our system, (x,y,z), by using determinants. If you need clarification on how to obtain the formulas below, please see the explanation at the bottom of the solution.
x=| cc 4 & 2 & 0 6 & 4 & 2 29 & 3 & 4 |/|C|
y=| cc 5 & 4 & 0 3 & 6 & 2 7 & 29 & 4 |/|C|
z=| cc 5 & 2 & 4 3 & 4 & 6 7 & 3 & 29 |/|C|Let's start by calculating |C|.
[ 5* 4* 4+ 2* 2* 7+ 0* 3* 3]
-
[ 7* 4* 0+ 3* 2* 5+ 4* 3* 2]
⇕
108-54=54
Let's substitute 54 for |C| in the corresponding formula to find the value of x.
In a similar way, we can find the values for y and z.
y-variable
z-variable
y=| cc 5 & 4 & 0 3 & 6 & 2 7 & 29 & 4 |/|C|
z=| cc 5 & 2 & 4 3 & 4 & 6 7 & 3 & 29 |/|C|
y=| ccc 5 & 4 & 0 3 & 6 & 2 7 & 29 & 4 |/54
z=| ccc 5 & 2 & 4 3 & 4 & 6 7 & 3 & 29 |/54
y=-162/54
z=324/54
y=-3
z=6
The solution of the system, which is the point of intersection of the planes, is (2,-3,6).
Extra
Obtaining the formulas for x, y, and z.
Consider our system.
5x+ 2y+ 0z=4 3x+ 4y+ 2z=6 7x+ 3y+ 4z=29
To find the value of the x-variable, we need to calculate the quotient between two determinants. As we have seen before, the divisor will be |C|, the determinant of the coefficient matrix. To find the dividend, substitute in C the result of each of the three equations for the coefficients of x. Then, calculate the determinant of the resulting matrix.
result column
↓
x=| cc 4 & 2 & 0 6 & 4 & 2 29 & 3 & 4 |/|C|
We find y and z in a similar way. In the coefficient matrix, substitute the results of the equations for the respective coefficients. Calculate the determinant of the resulting matrices and divide by the determinant of the coefficient matrix, |C|.
result column result column
↓ ↓
y=| cc 5 & 4 & 0 3 & 6 & 2 6 & 29 & 4 |/|C|
z=| cc 5 & 2 & 4 3 & 4 & 6 7 & 3 & 29 |/|C|
