Question

NO LINKS!!
Find the value of k such that the system of linear equations is inconsistent.
4x - 8y = -3
2x + ky = 10

k = _____

Answer 1

Answer:

k = -4

Step-by-step explanation:

Given system of linear equations:

$\begin{cases}4x - 8y = -3\\2x + ky = 10\end{cases}$

Rewrite the given equations so that they are in slope-intercept form.

$\begin{aligned}4x-8y&=-3\\4x&=8y-3\\4x+3&=8y\\\dfrac{4x+3}{8}&=y\\y&=\dfrac{1}{2}x+\dfrac{3}{8}\end{aligned}$

$\begin{aligned}2x+ky&=10\\ky&=-2x+10\\y&=\dfrac{-2x+10}{k}\\y&=-\dfrac{2}{k}x+\dfrac{10}{k}\end{aligned}$

An inconsistent system of linear equations has no solutions and therefore the lines are parallel.  

Parallel lines have the same slope:

$\implies \dfrac{1}{2}=-\dfrac{2}{k}$

$\implies \dfrac{1}{2}k=-2$

$\implies k=-2 \cdot 2$

$\implies k=-4$

Answer 2

Answer:

• The system is inconsistent if k = -4

--------------------------

• Inconsistent system is a system with no solutions.
• Parallel lines are inconsistent.
• Parallel lines have equal slopes but different y-intercepts.

Equation in the standard form ax + by = c  has:

• slope = - a/b,
• y-intercept =  c/b.

Given system:

• 4x - 8y = -3
• 2x + ky = 10

Compare the slopes of the two lines:

• - 4 / - 8 = 1/2 and
• - 2/k

We need them to be equal:

• 1/2  = - 2/k
• k = - 4

Compare the y-intercepts of the two lines:

• - 3/ - 8 = 3/8 and
• 10 / k

We need to confirm they are not same:

• 3/8 ≠ 10/k
• 3/8 ≠ 10/-4, correct as positive and negative numbers

Confirmed, so k = -4 is our solution.