Question

Find value(s) of k so that the linear system is consistent? (Enter your answers as a comma-separated list.) 8x1 − 3x2 = 2 12x1 + kx2 = −1

Answer 1

Answer:

For each value of $k \neq -4.5$, 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.

$8x_{1} - 3x_{2} = 2$

$8x_{1} = 2 + 3x_{2}$

$x_{1} = \frac{2 + 3x_{2}}{8}$

Replacing in the second equation:

$12x_{1} + kx_{2} = -1$

$12\frac{2 + 3x_{2}}{8} + kx_{2} = -1$

$3\frac{2 + 3x_{2}}{2} + kx_{2} = -1$

$3(2 + 3x_{2}) + 2kx_{2} = -2$

$6 + 9x_{2} + 2kx_{2} = -2$

$x_{2}(9 + 2k) = -8$

$x_{2} = -\frac{8}{9 + 2k}$

$x_{2}$ is not going to be consistet if it divided by 0. So we need:

$9 + 2k \neq 0$

$2k \neq -9$

$k \neq -4.5$