Systems of Equations Question 7

For what value of $a$ does the system have no solution? $$ -3x+y=6$$ $$ax+2y=4$$

$ \dfrac{1}{2} $

$-6$

$-3$

$3$

Approach

For a system of linear equations to have no solution, they need to have the same slope and different $y$-intercepts. Rearranging the two equations into slope-intercept form,

$$ -3x+y=6$$ $$ y=3x+6$$

$$ax+2y=4$$ $$2y=-ax+4$$ $$y=-\frac{a}{2}x+2$$

Setting the slopes equal to each other,

$$ 3=-\frac{a}{2} $$ $$\boxed{-6}=a$$