Q.Solve the following system of equations using Cramer's rule: $x - 2y + 3z = 1$, $2x + y - z = 3$ and $3x - y + 2z = -2$.

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.

The Precise Statement

A linear system (in two variables) is a collection of equations of the form:

where $a_1, b_1, c_1, a_2, b_2, c_2$ 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 $x^2$, no $xy$, no $\frac{1}{x}$.

A solution is an ordered pair $(x, y)$ that makes both equations true when substituted in.

The Three Possibilities

How to Solve: Two Core Methods

1. Substitution Method

Solve one equation for one variable, then plug that into the other equation.

From the apple-banana problem:

$\begin{cases} 2x + 3y = 17 \\ x + 2y = 10 \end{cases}$ …