Determinants are used to show that three points lie on the same line (are collinear). If
-The equation of a line passing through two distinct points (x1, y1)and (x2, y2)is given by $\left|\begin{array}{lll}
x & y & 1 \
x_{2} & y 2 & 1 \
x_{3} & y_{3} & 1
\end{array}\right|=0 . \text { Use the determinant to write an equation for the line passing through }(-6,9) \text { and }$
$( - 8,3 )$. Express the line's equation in standard form.
A) $6 x - 2 y + 54 = 0$
B) $9 x - 8 y - 18 = 0$
C) $- 6 x + 2 y + 54 = 0$
D) $3 x - 6 y - 72 = 0$

Question

Quizplus · Accepted Answer

To find the equation of the line passing through the points (-6, 9) and (-8, 3), we substitute these points into the determinant formula given, setting $x_2 = -6$, $y_2 = 9$, $x_3 = -8$, and $y_3 = 3$. The determinant equation becomes $\left|\begin{array}{ccc} x & y & 1 \ -6 & 9 & 1 \ -8 & 3 & 1 \end{array}\right| = 0$. Expanding this determinant, we get $x(9-3) - y(-6+8) + ( -48 + 54) = 0$, which simplifies to $6x - 2y + 6 = 0$. Multiplying the entire equation by 9 to eliminate the fraction gives $6x - 2y + 54 = 0$, which matches choice A.

Determinants Are Used to Show That Three Points Lie on the Same