10
algebra

What is the relationship between the lines? {10x15y=202x3y=4\begin{cases} 10x - 15y = 20 \\ 2x - 3y = 4 \end{cases}

A

The lines intersect at one point

B

The lines are parallel but not identical

C

The lines are perpendicular

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: Divide the first equation by 5: 10x15y5=205\frac{10x - 15y}{5} = \frac{20}{5}2x3y=42x - 3y = 4

This is exactly the second equation!

Conclusion: The lines are identical (coincident)

This means infinitely many solutions—every point on one line is also on the other.

💡 Strategic Tip: Identical lines are also called "coincident lines" or "the same line."

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

Choice B is incorrect because parallel but not identical would give no solution (different constants).

Choice C is incorrect because both lines have slope 23\frac{2}{3}, not perpendicular slopes.