Hi Mike… can you solve this and explain please? Thanks!

ax + 5y = 8
12x + 15y = 10

In the given system of equations, a is a constant. If the system has no solutions, what is the value of a ?

When a linear system has no solutions, that means the lines formed by the equations are parallel—they never intersect, so there is never a points that satisfies both equations. And of course, parallel lines have equal slopes.

So, let’s put both of these equations into slope-intercept form:

ax+5y=8\\5y=-ax+8\\\\y=-\dfrac{a}{5}x+\dfrac{8}{5}

12x+15y=10\\15y=-12x+10\\\\y=-\dfrac{12}{15}x+\dfrac{10}{15}\\\\y=-\dfrac{4}{5}x+\dfrac{2}{3}

All we care about are the slopes, so let’s set those equal and solve.

-\dfrac{a}{5}=-\dfrac{4}{5}\\\\a=4

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