Solving Linear Systems: From Intuition to Precision
Imagine you're at a market. You buy 2 apples and 3 bananas for ₹17. Your friend buys 1 apple and 2 bananas for ₹10. How much does one apple cost? One banana?
You already know how to solve this with a single variable — but here, two things are unknown at once. The price of an apple and the price of a banana are linked together by two different conditions. That's a linear system: a set of equations that must all be true at the same time.
The Intuition: Two Lines, One Meeting Point
Each equation in a system of two variables (say x and y) can be drawn as a straight line on a graph. The equation 2x+3y=17 is one line; x+2y=10 is another.
Solving the system means finding the (x,y) pair that lies on both lines at once. That's the point where the two lines cross — their intersection. If you think about it, that's the only place where both conditions are satisfied simultaneously.
Tip
Visualise it: two straight lines on a plane. They either cross at exactly one point (one solution), run parallel and never meet (no solution), or lie exactly on top of each other (infinitely many solutions). The first case is what you'll see most often.
The Precise Statement
A linear system (in two variables) is a collection of equations of the form:
{a1x+b1y=c1a2x+b2y=c2
where a1,b1,c1,a2,b2,c2 are known numbers (constants), and x and y are the unknowns we need to find. The word "linear" means each variable appears only to the first power — no x2, no xy, no x1.
A solution is an ordered pair (x,y) that makes both equations true when substituted in.
a1x+b1y=c1anda2x+b2y=c2
The Three Possibilities
Case
Graph
Number of Solutions
Lines intersect at one point
Two crossing lines
Exactly one
Lines are parallel (same slope, different intercept)
Two parallel lines
None
Lines are identical (same slope and intercept)
One line on top of another
Infinitely many
How to Solve: Two Core Methods
1. Substitution Method
Solve one equation for one variable, then plug that into the other equation.
Cramer's rule expresses each unknown as a ratio of determinants, x=Dx/D (and similarly for y, z), so each part below is solved by computing the coefficient determinant D and the numerator determinants; when D=0, the system has either infinitely many solutions (if every numerator determinant also vanishes) or no solution (if any does not). …
(iv)x−3y=4,3x−9y=12: D=13−3−9=0,Dx=412−3−9=0,Dy=13412=0. Since D=Dx=Dy=0 and the second equation is 3× the first, the system is consistent with infinitely many solutions: x=4+3y,y∈R.
(v)−2x+y=3,4x−2y=5: D=−241−2=0,Dx=351−2=−11=0. With D=0 but Dx=0 the system is inconsistent — no solution.