which equation can pair with 3x + 4y = 8 to create a consistent and independent system?\n6x + 8y = 16\n-3x…

Explanation:

Step1: Recall system - of - equations concepts

A consistent and independent system has exactly one solution. Two linear equations (a_1x + b_1y=c_1) and (a_2x + b_2y = c_2) form a consistent - independent system if (\frac{a_1}{a_2}\neq\frac{b_1}{b_2}). The given equation is (3x + 4y=8), where (a_1 = 3), (b_1 = 4), and (c_1 = 8).

Step2: Check each option

Option 1: (6x + 8y = 16)

Here, (a_2=6), (b_2 = 8), (\frac{a_1}{a_2}=\frac{3}{6}=\frac{1}{2}), (\frac{b_1}{b_2}=\frac{4}{8}=\frac{1}{2}). Since (\frac{a_1}{a_2}=\frac{b_1}{b_2}), this is a dependent system.

Option 2: (-3x-4y=-6)

Here, (a_2=-3), (b_2=-4), (\frac{a_1}{a_2}=\frac{3}{-3}=-1), (\frac{b_1}{b_2}=\frac{4}{-4}=-1). Since (\frac{a_1}{a_2}=\frac{b_1}{b_2}), this is a dependent system.

Option 3: (6x - 3y=2)

Here, (a_2 = 6), (b_2=-3), (\frac{a_1}{a_2}=\frac{3}{6}=\frac{1}{2}), (\frac{b_1}{b_2}=\frac{4}{-3}=-\frac{4}{3}). Since (\frac{a_1}{a_2}\neq\frac{b_1}{b_2}), this equation forms a consistent and independent system with (3x + 4y = 8).

Option 4: (-3x-4y=-8)

Here, (a_2=-3), (b_2=-4), (\frac{a_1}{a_2}=\frac{3}{-3}=-1), (\frac{b_1}{b_2}=\frac{4}{-4}=-1). Since (\frac{a_1}{a_2}=\frac{b_1}{b_2}), this is a dependent system.

Answer:

C. (6x - 3y = 2)