Answer:
a = 6
Explanation:
If the system of equations has no solution, their graphs have the same slope.
1. Slope of first equation
[tex]\begin{array}{rcl}|\frac{1}{2}x - \frac{2}{3} y|& = & 7\\|3x - 4y| & = & 42\\3x - 4y & = & \pm42\\-4y & = & -3x \pm 42\\y & = & \frac{3}{4}x \mp \frac{42}{4}\\\\\text{Slope}& = & \dfrac{3}{4} \end{array}[/tex]
2. Slope of second equation
[tex]\begin{array}{rcl}|ax - 8y| & = & 1\\ax - 8y & = & \pm 1\\-8y & = & -ax \pm 1\\y & = & \frac{a}{8}x\mp \frac{1}{8}\\\\\text{Slope} & = & \dfrac{a}{8}\\\end{array}[/tex]
3. Value of a
[tex]\begin{array}{rcl}\dfrac{3}{4} & = & \dfrac{a}{8}\\\\a & = & \dfrac{24}{4}\\\\ & = & \mathbf{6}\\\end{array}[/tex]
Check:
[tex]\begin{array}{rcrrcl}|\frac{1}{2}x - \frac{2}{3} y|& = & 7 & \qquad |6x - 8y| & = &1\\|3x - 4y| & = & 42 &|3x - 4y| &= &\frac{1}{2} & \\\end{array}\\\text{Both equations have the same slope}[/tex]
