How this instrument works
Given two lines in slope-intercept form, y = m₁x + b₁ and y = m₂x + b₂, the point where they cross is the single (x, y) pair that satisfies both equations at once. Setting the two right-hand sides equal, m₁x + b₁ = m₂x + b₂, and solving for x directly gives x = (b₂−b₁) ⁄ (m₁−m₂); substituting that x back into either original equation then gives y.
This is the same underlying problem the more general Systems of Equations page elsewhere on this site solves via Cramer's Rule, but starting from slope-intercept form specifically rather than standard form — a common starting point whenever two lines are already described the way they'd be graphed, by their slope and y-intercept, rather than by standard-form coefficients.
The denominator, m₁−m₂, reveals when a genuine crossing point fails to exist: if the two slopes are equal, the lines are parallel and never meet (unless they're actually the identical line, sharing infinitely many points), and this calculator flags that case rather than dividing by zero.
- Enter the first line's slope and y-intercept into the Line 1 fields.
- Enter the second line's slope and y-intercept into the Line 2 fields.
- Read Intersection: x and Intersection: y: the sheet solves for the single point where both lines cross.
Worked example — y = 2x+1 and y = −x+10
The lines y = 2x + 1 and y = −x + 10 cross at x = (10−1) ⁄ (2−(−1)) = 9 ⁄ 3 = 3, and y = 2(3)+1 = 7 — the point (3, 7). Checking against the second equation confirms it too: −(3)+10 = 7, matching exactly.
The lines y = x and y = −x + 4 cross at x = (4−0) ⁄ (1−(−1)) = 4⁄2 = 2, and y = 2 — the point (2, 2), a simple case where both lines carry unit-magnitude slopes of opposite sign, meeting exactly halfway between their two y-intercepts.
Questions
How do you find where two lines intersect?
Set the two lines' equations equal to each other (since both must give the same y at the crossing point), solve for x, then substitute that x back into either original equation to find y — the resulting (x, y) pair satisfies both lines simultaneously.
What happens if the two lines have the same slope?
They're parallel and never cross (unless their y-intercepts also match, making them the identical line with infinitely many shared points) — the intersection formula's denominator, m₁−m₂, becomes zero, and this calculator flags that case rather than dividing by zero.
How is this different from the general Systems of Equations page?
That page solves a system given in standard form, a₁x+b₁y=c₁ and a₂x+b₂y=c₂, using Cramer's Rule. This page instead starts from slope-intercept form, y=mx+b, the more natural starting point when two lines are already described the way they'd be graphed.
Can the intersection point have negative coordinates?
Yes — depending on the two lines' slopes and y-intercepts, the crossing point can land anywhere on the coordinate plane, including negative x, negative y, or both, with no special handling needed.
What does the intersection point represent physically?
It's the one combination of x and y values where both linear relationships hold true simultaneously — in a practical context, this might be a break-even point, a point where two rates of change meet, or simply where two graphed lines visibly cross.