Question

If the system of linear equations below has no solution, and a is a constant, what is the value of a?"A. -2 B. -½ C. 2 D. and 2 equations in system are absolute value 1/2 x - 2/3 y = 7 and absolute value of ax-8y= -1

Answer 1

a = 6

Step-by-step explanation:

If the system of equations has no solution, their graphs have the same slope.

1. Slope of first equation

`\begin{array}{rcl}|(1)/(2)x - (2)/(3) y|& = & 7\\|3x - 4y| & = & 42\\3x - 4y & = & \pm42\\-4y & = & -3x \pm 42\\y & = & (3)/(4)x \mp (42)/(4)\\\\\text{Slope}& = & (3)/(4) \end{array}`

2. Slope of second equation

`\begin{array}{rcl}|ax - 8y| & = & 1\\ax - 8y & = & \pm 1\\-8y & = & -ax \pm 1\\y & = & (a)/(8)x\mp (1)/(8)\\\\\text{Slope} & = & (a)/(8)\\\end{array}`

3. Value of a

`\begin{array}{rcl}(3)/(4) & = & (a)/(8)\\\\a & = & (24)/(4)\\\\ & = & \mathbf{6}\\\end{array}`

Check:

`\begin{array}{rcrrcl}|(1)/(2)x - (2)/(3) y|& = & 7 & \qquad |6x - 8y| & = &1\\|3x - 4y| & = & 42 &|3x - 4y| &= &(1)/(2) & \\\end{array}\\\text{Both equations have the same slope}`