solve the system of equation using a matrix

Question

1.5x - 0.5y = -3.5
1.5x + 1.5y = -7.5

a) no solution
b)[-2 -3]
c)[-3 -2]
d)[3 2]

Answer 1

To solve the system of equations using a matrix, we can write the system in matrix form:

[1.5 -0.5 | -3.5]
[1.5 1.5 | -7.5]

We can solve this system by performing row operations to put the matrix in row-echelon form (also known as Gaussian elimination).

First, let's multiply the first equation by 2 and add it to the second equation to eliminate x:

2(1.5x - 0.5y) = 2(-3.5)
3x - y = -7

Now our system becomes:

[1.5 -0.5 | -3.5]
[3 -1 | -7 ]

Next, let's multiply the first row by 2/3 to simplify:

(2/3)(1.5x - 0.5y) = (2/3)(-3.5)
x - (1/3)y = -7/3

Now our system becomes:

[1 -1/3 | -7/3]
[3 -1 | -7 ]

To eliminate x in the second row, we can multiply the first row by -3 and add it to the second row:

-3(1 -1/3 | -7/3) = -3(-7/3)
-3x + y = 7

Now our system becomes:

[1 -1/3 | -7/3]
[0 -2/3 | 14/3]

Finally, let's multiply the second row by -3/2 to simplify:

(3/2)(0 -2/3 | 14/3) = (3/2)(14/3)
0 1 | -7

Now our system becomes:

[1 -1/3 | -7/3]
[0 1 | -7 ]

We have successfully transformed the matrix to row-echelon form. Now we can solve the system by back substitution.

From the second row, we can conclude that y = -7.

Substituting this value of y into the first row, we get:

x - (1/3)(-7) = -7/3
x + 7/3 = -7/3
x = -7/3 - 7/3
x = -14/3

Therefore, the solution to the system is x = -14/3 and y = -7.

In matrix form, this can be written as:

[-14/3 -7]

Hence, the correct answer is option d) [-14/3 -7].