# Matrices Word Problems

Matrices Word Problems
• Page 1
1.
Solve system of equations. Determine whether the system is consistent and independent, consistent and dependent, or inconsistent by using augmented matrices.
2$x$ + 5$y$ = 11
4$x$ - 3$y$ = 9
 a. (1, 3) b. (3, 1) c. (- 1, 4) d. ($\frac{1}{2}$, 2)

#### Solution:

 (25|114- 3|9) R1R2

[Write the augmented matrix.]

Apply the row operation 1 / 2 × R1 to get 1 in the first row, first column.
(152|1124- 3|9)

Apply the row operation - 4 × R1 + R2 to get 0 in the second row, first column.
(152|1120- 13|- 13)

Apply the row operation - 1 / 13 × R2 to get 1 in the second row, second column.
(152|11201|1)

Apply the row operation -5 / 2 × R2 + R1 to get 0 in the first row, second column.
(10|301|1)

So, x = 3, y = 1
The solution is (3, 1).

2.
Choose the augmented matrix for the following system of equations.
6$x$ + 4$y$ - $z$ = 4
$x$ + $y$ + $z$ = 2
4$x$ - $y$ + 6$z$ = 4
 a. $\left(\begin{array}{cccc}4& - 1& 6& |4\\ 1& 1& 1& |2\\ 6& 4& - 1& |4\end{array}\right)$ b. $\left(\begin{array}{cccc}1& 1& 1& |4\\ 6& 4& - 1& |2\\ 4& - 1& 6& |4\end{array}\right)$ c. $\left(\begin{array}{cccc}6& 4& - 1& |4\\ 1& 1& 1& |2\\ 4& - 1& 6& |4\end{array}\right)$ d. $\left(\begin{array}{cccc}6& 1& - 1& |4\\ 2& 1& 1& |2\\ 4& - 1& 6& |4\end{array}\right)$

#### Solution:

The augmented matrix for the system of equations in the question is
(64- 1|4111|24- 16|4)

3.
Which one is the augmented matrix for the system of equations?
$x$ - 2$y$ + $z$ = 4
5$x$ - $z$ = 4
$y$ + 5$z$ = - 7
 a. $\left(\begin{array}{cccc}1& 1& - 2& |4\\ 5& 0& - 1& |4\\ 0& 1& 5& |- 7\end{array}\right)$ b. $\left(\begin{array}{ccc}1& - 2& 1\\ 0& 5& - 1\\ 0& 1& 5\end{array}\right)$ c. $\left(\begin{array}{cccc}1& - 2& 1& |4\\ 5& 0& - 1& |4\\ 0& 1& 5& |- 7\end{array}\right)$ d. $\left(\begin{array}{ccc}1& - 2& 1\\ 5& 0& - 1\\ 1& 0& 5\end{array}\right)$

#### Solution:

The augmented matrix for the system of equations is:
(1- 21|450- 1|4015|- 7)
[The coefficients are seperated from the constants by the vertical line segment.]

4.
Solve system of equations. Determine whether the system is consistent and independent, consistent and dependent, or inconsistent by using augmented matrices.
$x$ - $y$ + 3$z$ = 5
4$x$ + 2$y$ - $z$ = 0
$x$ +3$y$ + $z$ = 5
 a. (10, - 10, - 5); consistent and independent b. (0, 1, 2); consistent and independent c. (10, - 10, - 5); consistent and dependent d. no solution; inconsistent

#### Solution:

 (1-13|542-1|0131|5) R1R2R3

[Write the augmented matrix.]

Apply the row operation - 4×R1 + R2 to get 0 in the second row, first column.
(1-13|506-13|-20131|5)

Apply the row operation -1×R1 + R3 to get 0 in the third row, first column.
(1-13|506-13|-2004-2|0)

Apply the row operation 1 / 6×R2 to get 1 in the second row, second column.
(1-13|501-136|-20604-2|0)

Apply the row operation R2 + R1 to get 0 in the first row, second column.
(1056|10601-136|-20604-2|0)

Apply the row operation - 4×R2 + R3 to get 0 in the third row, second column.
(1056|10601-136|-20600203|403)

Apply the row operation 3 / 20×R3 to get 1 in the third row, third column.
(1056|10601-136|-206001|2)

Apply the row operation - 5 / 6×R3 + R1 to get 0 in the first row, third column.
(100|001-136|-206001|2)

Apply the row operation 13 / 6×R3 + R2 to get 0 in the second row, third column.
(100|0010|1001|2)

So, x = 0, y = 1, z = 2

So, the system is consistent and independent.

5.
Identify the augmented matrix for the system of equations.
$x$ + $z$ = 8
$y$ - $z$ = 8
$x$ + $y$ = 6
 a. $\left(\begin{array}{cccc}1& 0& 1& |8\\ 0& 1& - 1& |- 8\\ 1& 1& 1& |6\end{array}\right)$ b. $\left(\begin{array}{cccc}1& 0& 1& |8\\ 0& 1& - 1& |8\\ 1& 1& 0& |6\end{array}\right)$ c. $\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& - 1\\ 1& 1& 0\end{array}\right)$ d. $\left(\begin{array}{cccc}1& 1& 1& |8\\ 1& 1& - 1& |8\\ 1& 1& 0& |6\end{array}\right)$

#### Solution:

The augmented matrix for the system of equations given in the question is:
(101|801- 1|8110|6)
[The coefficients are separated from the constants by the vertical line segment.]

6.
Identify the augmented matrix for the system of equations.
$\frac{3}{2}x+\frac{1}{2}y$ = $z$ - 3
= 6
$x$ - $\frac{1}{3}z$ = - 6
 a. $\left(\begin{array}{cccc}\frac{3}{2}& 1& 1& |- 3\\ \frac{1}{3}& - 1& \frac{1}{2}& |6\\ 1& 0& \frac{\mathrm{- 1}}{3}& |6\end{array}\right)$ b. c. $\left(\begin{array}{ccc}3& 1& 0\\ 1& 1& 2\end{array}\right)$ d. None of the above

#### Solution:

The augmented matrix for the system of equations given in the question is:
(3212- 1|- 313- 112|  610- 13|- 6)
[The coefficients are seperated from the constants by the vertical line segment.]

7.
Solve the system of equations.
$x$ + $z$ = 10
$y$ + $z$ = 7
$x$ + $y$ = 9
Determine whether the system is consistent and independent, consistent and dependent, or inconsistent by using augmented matrices.
 a. (6, 3, 4); consistent and independent b. (7, 2, 4); consistent and independent c. (7, 2, 4); consistent and dependent d. (6, 2, 5); inconsistent

#### Solution:

 (101|10011|7110|9) R1R2R3

[Write the augmented matrix.]

Apply the row operation - 1 × R1 + R3 to get 0 in the third row, first column.
(101|10011|701- 1|-1)

Apply the row operation - 1 × R2 + R3 to get 0 in the third row, second column.
(101|10011|700- 2|-8)

Apply the row operation - 1 / 2 × R3 to get 1 in the third row, third column.
(101|10011|7001|4)

Apply the row operation - 1 × R3 + R1 to get 0 in the first row, third column.
(100|6011|7001|4)

Apply the row operation - 1 × R3 + R2 to get 0 in the second row, third column.
(100|6010|3001|4)

So, x = 6, y = 3, z = 4
The solution is (6, 3, 4).

So, the system is consistent and independent.

8.
Which of the following represents the system of equations for the augmented matrix?

 a. 7$x$ - $y$ = - 6, 5$x$ + $y$ = - 5 b. 7$x$ - $y$ = 6, 5$x$ + $y$ = 5 c. 7$x$ + $y$ = 6, 5$x$ - $y$ = - 5 d. 7$x$ - $y$ = 6, 5$x$ + $y$ = - 5

#### Solution:

(7- 15   1) (xy) = (   6- 5)

(7x-y5x+y) = (   6- 5)
[Multiplication of matrices.]

7x - y = 6, 5x + y = - 5
[By the definition of equal matrices.]

9.
Find the system of equations represented by the augmented matrix.

 a. IV b. II c. I d. III

#### Solution:

(4- 19|- 4- 101|   49101|   8)

(4x-y+10z- x+0y+z9x+10y+z) = (- 4   4   8)
[Multiplication of matrices.]

4x - y + 9z = - 4
- x + z = 4
9x + 10y + z = 8
[By the definition of equal matrices.]

10.
Find the number of solutions that the augmented matrix has. $\left(\begin{array}{cccc}1& 0& 0& |2\\ 0& 1& 0& |9\\ 0& 0& 1& |7\end{array}\right)$
 a. infinite b. 1 c. 2

#### Solution:

(100010001) (xyz) = (297)

(1x+0y+0z0x+1y+0z0x+0y+1z) = (297)
[Multiplication of matrices.]

(xyz) = (297)

x = 2, y = 9 & z = 7.
[By the definition of equal matrices.]

So, the number of solutions of the system of equations represented by the augmented matrix is one.