Math
Intersection of Two Lines Calculator
Enter two lines in any form (slope-intercept, two points, point and slope, standard, or general) and mathcheck finds their intersection point as exact fractions.
It tells you when the lines are parallel or the same line, reports the angle between them, flags perpendicular pairs, graphs both lines, and shows the solving steps.
Type a fraction like 3/2 or a decimal like -0.5. Type "undefined" for a vertical line.
Where the line crosses the y-axis.
Type a fraction like 3/2 or a decimal like -0.5. Type "undefined" for a vertical line.
Where the line crosses the y-axis.
Try an example
Result
Intersection point
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- Relationship
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- Angle between the lines
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- Line 1
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- Line 2
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Study path
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What the intersection of two lines means
The intersection point is the one (x, y) pair that satisfies both line equations at the same time; it is the solution of the two-equation system. Two distinct non-parallel lines always meet at exactly one point. Parallel lines never meet (no solution), and two equations that describe the same line meet everywhere (infinitely many solutions).
How to tell what kind of pair you have
| Slopes | Intercepts | Result |
|---|---|---|
| Different | Any | One intersection point |
| Same | Different | Parallel: no intersection |
| Same | Same | Coincident: the same line, infinitely many points |
| m1 × m2 = -1 | Any | Intersect at a right angle (perpendicular) |
y = x + 1 and y = -x + 5
Set them equal: x + 1 = -x + 5, so 2x = 4 and x = 2. Then y = 2 + 1 = 3. The lines meet at (2, 3), and since 1 × (-1) = -1 they are perpendicular.
How to use this calculator
- For each line, choose the form you have and enter its numbers. The two lines do not need to use the same form.
- Read the intersection point in the header as exact fractions, with the decimal version beside it.
- Check the relationship badge: intersecting, perpendicular, parallel, or coincident.
- Use the graph to confirm the crossing visually and the angle value if the problem asks for it.
- Open Show the work for the One number computed from a square matrix that tells you whether it can be inverted. and Cramer's-rule steps you can copy into homework.
How to read the answer
An intersection point such as (12/7, -1/14) is the single (x, y) pair that makes both equations true; plug it into either equation to verify. If the calculator says parallel, the system has no solution, and if it says coincident, every point on the line is a solution. The angle is the acute angle between the lines; 90° means the lines are perpendicular.
Common mistakes and edge cases
- Solving for x and forgetting to find y. The intersection needs both coordinates.
- Arithmetic slips when the How steep a line is: how much y changes for each step right in x. are fractions. Keep everything as fractions until the end; the calculator does.
- Assuming two lines with the same slope intersect. Check the A point where a graph crosses an axis.: same slope means parallel or coincident.
- Entering the same line twice by accident (for example y = 2x + 1 and 2x - y = -1). The calculator reports coincident lines.
- Reading the angle as the obtuse angle. Lines form two supplementary angles; the calculator reports the acute one.
Worked examples
Perpendicular pair
y = x + 1 and y = -x + 5
Intersection point
(2, 3)
Fraction intersection from standard forms
3x + 2y = 5 and x - 4y = 2
Intersection point
(12/7, -1/14)
Mixed forms
Line through (0, 0) and (4, 2), and y = -2x + 5
Intersection point
(2, 1)
Vertical line
x = 3 and y = 2x - 1
Intersection point
(3, 5)
Parallel lines
y = 2x + 1 and y = 2x - 4
Intersection point
none (parallel lines)
Same line twice
y = 2x + 1 and 2x - y + 1 = 0
Intersection point
every point (same line)
Decimal coefficients
y = 0.5x + 1.25 and y = -1.5x + 3.25
Intersection point
(1, 7/4)
Invalid line
Line 1 given by two identical points
Intersection point
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Frequently asked questions
How do I find where two lines intersect?+
Set the two equations equal (or use elimination), solve for x, then substitute back to get y. For y = m1x + b1 and y = m2x + b2, x = (b2 - b1)/(m1 - m2).
What if the two lines have the same slope?+
They are either parallel (different intercepts, no intersection) or the same line (identical equations, infinitely many intersection points). The calculator tells you which.
How is the angle between two lines calculated?+
tan θ = |(m2 - m1)/(1 + m1·m2)|. The calculator uses the lines' normal vectors instead so it also works for vertical lines, and it reports the acute angle between 0° and 90°.
Is the intersection point the solution of the system of equations?+
Yes. Each line is one linear equation, and their intersection is the (x, y) pair that satisfies both, which is exactly the solution of the 2×2 system.
Can I enter the lines in different forms?+
Yes. Each line has its own form selector, so you can give one line as two points and the other in standard form.
Why is the answer a fraction?+
Intersection coordinates are often fractions, such as (12/7, -1/14). Fractions are exact and will check perfectly in both equations; the decimal version is shown for convenience.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026