10
algebra

What is the relationship between the lines? {2x3y=94x6y=18\begin{cases} 2x - 3y = 9 \\ 4x - 6y = 18 \end{cases}

A

The lines are perpendicular

B

The lines are parallel but not identical

C

The lines intersect at one point

D

The lines are identical

Correct Answer: D

Choice D is the correct answer. Let's check if one equation is a multiple of the other.

Step 1: Multiply the first equation by 2: 2(2x3y)=2(9)2(2x - 3y) = 2(9)4x6y=184x - 6y = 18

This is exactly the second equation!

Step 2: Both equations represent the same line → identical (coincident) lines

This means infinitely many solutions.

💡 Strategic Tip: Identical lines are also called "coincident lines" and always give infinitely many solutions.

Choice A is incorrect because perpendicular lines have slopes whose product is -1. Here, both have slope 23\frac{2}{3}.

Choice B is incorrect because the lines are not just parallel—they're the same line.

Choice C is incorrect because intersecting at one point requires different slopes.