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

• The system is inconsistent if k = -4

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• 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.

Answer 2

k = -4

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\\(4x+3)/(8)&=y\\y&=(1)/(2)x+(3)/(8)\end{aligned}`

`\begin{aligned}2x+ky&=10\\ky&=-2x+10\\y&=(-2x+10)/(k)\\y&=-(2)/(k)x+(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 (1)/(2)=-(2)/(k)`

`\implies (1)/(2)k=-2`

`\implies k=-2 \cdot 2`

`\implies k=-4`