Q. For how many values of 'k', does the
following systems of equations have
infinitely many solutions
2x +y - 4z=k
4x + 3y - 12z = 5
x + 2y - 8z = 7
finitely many
0 1
O infinitely many
Answers
Step-by-step explanation:
Writing the Augmented Matrix of a System of Equations
A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.
For example, consider the following 2 × 2system of equations.
3x + 4y = 7
4x − 2y = 5
We can write this system as an augmented matrix (please note that a more common formatting for augmented matrices has a solid vertical line running through the matrix, rather than a line on each row):
[
3
4
|
7
4
−
2
|
5
]
We can also write a matrix containing just the coefficients. This is called the
coefficient matrix.
[
3
4
4
−
2
]
A three-by-three system of equations such as
3x − y − z = 0
x + y = 5
2x − 3z = 2
has a coefficient matrix
⎡
⎢
⎣
3
−
1
−
1
1
1
0
2
0
−
3
⎤
⎥
⎦
and is represented by the augmented matrix
⎡
⎢
⎣
3
−
1
−
1
∣
0
1
1
0
∣
5
2
0
−
3
∣
2
⎤
⎥
⎦
Notice that the matrix is written so that the variables line up in their own columns:
x-terms go in the first column, y-terms in the second column, and z-terms in the third column. It is very important that each equation is written ax + by + cz = d in standard form so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0.
How To
Given a system of equations, write an augmented matrix.
Write the coefficients of the x-terms as the numbers down the first column.
Write the coefficients of the y-terms as the numbers down the second column.
If there are z-terms, write the coefficients as the numbers down the third column.
Draw a vertical line and write the constants to the right of the line.
Example 1
Write the augmented matrix for the given system of equations.
x + 2y − z = 3
2x − y + 2z = 6
x − 3y + 3z = 4
2. Write the augmented matrix of the given system of equations.
4x − 3y = 11
3x + 2y = 4
Solutions
The augmented matrix displays the coefficients of the variables, and an additional column for the constants.
⎡
⎢
⎣
1
2
−
1
∣
3
2
−
1
2
∣
6
1
−
3
3
∣
4
⎤
⎥
⎦
[
4
−
3
∣
11
3
2
∣
4