Respuesta :
Answer:
For each value of [tex]k \neq -4.5[/tex], the system is consistent.
Step-by-step explanation:
A system is said to be consistent if it has at least one answer, that is, one set of values which satisfies the equations.
In this problem, we have that:
8x1 - 3x2 = 2
12x1 + kx2 = -1
I am going to write x1 as a function of x2 in the first equation, and then replace in the second.
[tex]8x_{1} - 3x_{2} = 2[/tex]
[tex]8x_{1} = 2 + 3x_{2}[/tex]
[tex]x_{1} = \frac{2 + 3x_{2}}{8}[/tex]
Replacing in the second equation:
[tex]12x_{1} + kx_{2} = -1[/tex]
[tex]12\frac{2 + 3x_{2}}{8} + kx_{2} = -1[/tex]
[tex]3\frac{2 + 3x_{2}}{2} + kx_{2} = -1[/tex]
[tex]3(2 + 3x_{2}) + 2kx_{2} = -2[/tex]
[tex]6 + 9x_{2} + 2kx_{2} = -2[/tex]
[tex]x_{2}(9 + 2k) = -8[/tex]
[tex]x_{2} = -\frac{8}{9 + 2k}[/tex]
[tex]x_{2}[/tex] is not going to be consistet if it divided by 0. So we need:
[tex]9 + 2k \neq 0[/tex]
[tex]2k \neq -9[/tex]
[tex]k \neq -4.5[/tex]
The value of k so that the linear system is consistent is [tex]k \ne - 4.5[/tex]
The equations are given as:
[tex]8x_1 - 3x_2 = 2[/tex]
[tex]12x_1 + kx_2 = -1[/tex]
Make x1 the subject in [tex]8x_1 - 3x_2 = 2[/tex]
[tex]x_1 = \frac{2 + 3x_2}{8}[/tex]
Substitute [tex]x_1 = \frac{2 + 3x_2}{8}[/tex] in the second equation
[tex]12 \times \frac{2 + 3x_2}{8} + kx_2 = -1[/tex]
Multiply through by 8
[tex]12 \times (2 + 3x_2) + 8kx_2 = -8[/tex]
Expand
[tex]24 + 36x_2 + 8kx_2 = -8[/tex]
Collect like terms
[tex]36x_2 + 8kx_2 = -8 - 24[/tex]
[tex]36x_2 + 8kx_2 = -32[/tex]
Factor out x2
[tex]x_2(36 + 8k) = -32[/tex]
Make x2 the subject
[tex]x_2 = -\frac{32}{36 + 8k}[/tex]
The equation is consistent if the denominator does not equal 0.
So, we have:
[tex]36 + 8k \ne 0[/tex]
Collect like terms
[tex]8k \ne - 36[/tex]
Divide through by 8
[tex]k \ne - 4.5[/tex]
Hence, the value of k so that the linear system is consistent is [tex]k \ne - 4.5[/tex]
Read more about consistent equations at:
https://brainly.com/question/13729904
