Question

which of the following statements is not true? choose the incorrect statement below. a. given any two distinct non - vertical lines in the cartesian plane, whether the lines are parallel to each other, perpendicular to each other, or neither can be determined by evaluating their slopes. b. if two non - vertical lines are perpendicular, then the product of their slopes must be negative one. c. an equation of the line through the point (-2,4) and perpendicular to y = 3x - 5 is x + 3y = 10. d. the lines y=\frac{3}{2}x + 7 and 2x - 3y = 5 are perpendicular.

Explanation:

• Option A: The relationship (parallel, perpendicular, or neither) between two non - vertical lines in the Cartesian plane can be determined by their slopes. If the slopes are equal, the lines are parallel. If the product of the slopes is - 1, the lines are perpendicular. So, this statement is true.
• Option B: For two non - vertical perpendicular lines, the product of their slopes is indeed - 1. This is a well - known property in coordinate geometry. So, this statement is true.
• Option C: The slope of the line $y = 3x-5$ is $m_1 = 3$. The slope of a line perpendicular to it is $m_2=-\frac{1}{3}$. Using the point - slope form $y - y_1=m(x - x_1)$ with the point $(-2,4)$ gives $y - 4=-\frac{1}{3}(x + 2)$. Rearranging, we get $x+3y = 10$. So, this statement is true.
• Option D: For the line $y=\frac{3}{2}x + 7$, the slope $m_1=\frac{3}{2}$. For the line $2x-3y = 5$, we can rewrite it in slope - intercept form $y=\frac{2}{3}x-\frac{5}{3}$, and its slope $m_2=\frac{2}{3}$. The product of the slopes $m_1\times m_2=\frac{3}{2}\times\frac{2}{3}=1

eq - 1$. So, these two lines are not perpendicular, and this statement is false.

Answer:

D. The lines $y=\frac{3}{2}x + 7$ and $2x - 3y = 5$ are perpendicular.