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Pair of Linear Equations: Intersection Point

Linear Equations · Class X

Two lines a1 x + b1 y = c1 and a2 x + b2 y = c2 , see where they meet (or don't).

2
type a value
-55
3
type a value
-55
4
type a value
-55
-1
type a value
-55
Live values
  • Determinant D-14
  • x at intersection (c1 = 6, c2 = 4)1.2857
  • y at intersection1.1429
  • Lines parallel? (D = 0)0
xy
  • Line 1: a1 x + b1 y = 6
  • Line 2: a2 x + b2 y = 4

Formulas in this lab

  • Determinant D
    a1b2−a2b1a_1 b_2 - a_2 b_1
  • x at intersection (c1 = 6, c2 = 4)
    (c1b2−c2b1)/D(c_1 b_2 - c_2 b_1)/D
  • y at intersection
    (a1c2−a2c1)/D(a_1 c_2 - a_2 c_1)/D
  • Lines parallel? (D = 0)
    D=0?D = 0?
Tip: When the determinant is zero, the lines are either parallel (no solution) or identical (infinite solutions).

Frequently asked questions

▶When do two lines intersect, coincide or stay parallel?

For $a_1 x + b_1 y = c_1$ and $a_2 x + b_2 y = c_2$: if $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$, lines meet at one point; if all three ratios are equal, they coincide (infinite solutions); if only the first two are equal but $\frac{c_1}{c_2}$ differs, they are parallel (no solution).

▶How do I use this lab?

Slide a1, b1, c1 and a2, b2, c2. The lab draws both lines and labels their intersection (or shows 'parallel'/'coincident'). Try $2x + 3y = 6$ and $4x + 6y = 12$ to see coincident lines.

▶Common mistake on this topic

Students check only $\frac{a_1}{a_2} = \frac{b_1}{b_2}$ and call the lines parallel, forgetting to compare with $\frac{c_1}{c_2}$. Without that third check, you can't tell parallel from coincident.

▶Where do we apply this?

If one shop sells 2 kg rice and 3 kg dal for Rs 600 and another sells 4 kg rice and 6 kg dal for Rs 1200, the equations coincide - both deals give the same per-kg pricing. Pair-of-equation thinking helps spot fair deals.