What is Linear system? — Mathematics

Definition 7.7 High School Mathematics · Chapter 7 — Equations of Lines and Linear Systems

A system of two linear equations in the unknowns $x$ and $y$ is a pair of equations

\begin{cases} a x + b y = c \\ a' x + b' y = c' \end{cases}

to be satisfied simultaneously. A solution is a pair $(x, y)$ satisfying both. Each equation describes a line, so solving the system means finding the intersection of two lines.

A linear system seen geometrically: each equation is a line, and the solution (2,1) of is their intersection point. — A linear system seen geometrically: each equation is a line, and the solution $(2,1)$ of Example 7.11 is their intersection point.

Examples

Example 7.13 (Modeling with a system)

Three notebooks and two pens cost $8$ ; one notebook and four pens cost $6$ . Let $x$ be the price of a notebook and $y$ that of a pen:

\begin{cases} 3x + 2y = 8 \\ x + 4y = 6 . \end{cases}

From the second equation, $x = 6 - 4y$ ; substituting, $3(6 - 4y) + 2y = 8$ , so $18 - 10y = 8$ , giving $y = 1$ and then $x = 2$ . A notebook costs $2$ and a pen costs $1$ .

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