Mr. IntegralCoincident Lines
Recognize different equations describing the same line.
Builds on Relative Positions of Two Lines · Parallel Lines
The bigger question: How can a picture become an equation?
On this page
Idea
Two equations can look completely different and still describe the same line.
Take and multiply every term by . You get . The equation looks different, but the graph is identical. These are called coincident lines: not two lines, but one line written twice.
Rule
Two lines are coincident exactly when every coefficient ratio agrees:
Equivalently, one equation is a non-zero multiple of the other. They share every one of their infinitely many points.
How it is used
Coincident is the third of the three positions. Only the constant separates it from parallel.
| All three ratios | Position |
|---|---|
| first two equal, third different | parallel - no shared points |
| all three equal | coincident - every point shared |
The fastest test is to ask whether one equation is a multiple of the other.
Divide the first coefficients to find the multiplier. Then check that the same multiplier works for the other two terms. If it works for all of them, the lines are coincident. If it works for the first two but not the constant, they are parallel.
Watch for a system with infinitely many solutions. That is what coincident lines look like in algebra.
It means the second equation told you nothing new. Two coincident lines give you one equation of information, not two. That is why such a system never has a single answer.
Worked example
Show that and are coincident. Divide the first coefficients to find the candidate multiplier: . Check the second coefficients carry the same ratio: , which matches. Check the constants: , which matches as well. All three ratios agree, so the second equation is the first multiplied by :
Explore
Getting the verdict to say Coincident takes care. Drag C and D until both sit exactly on the blue line. The purple line disappears underneath it. The meeting point stops naming one point and says every point instead.
Try this. Use A(0,0), B(1,1), C(0,1), D(1,2) for parallel lines. Change D to (1,0) for perpendicular lines.
Try it yourself.
Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.
Hint 1 · Find a starting point
See whether every coefficient, including the constant, has the same scale factor.
Hint 2 · Take the next step
The second equation is twice the first.
Show the reasoning
Answer: They are the same line
Multiplying an equation by a nonzero constant leaves its solution set unchanged. Every point of one line belongs to the other.
Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.
Pause before the next idea.
Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.