Meaning of a point satisfying a system of linear equations

📚 Definition of Satisfying a System of Linear Equations

In mathematics, a point satisfies a system of linear equations if, and only if, the coordinates of the point make all the equations in the system true. In simpler terms, when you substitute the $x$ and $y$ values of the point into each equation, both sides of every equation must be equal.

📜 Historical Context

The study of linear equations dates back to ancient civilizations, with early examples found in Babylonian and Egyptian mathematics. However, the systematic approach to solving systems of linear equations, including the concept of a point satisfying a system, developed more formally during the 17th and 18th centuries with advancements in algebra and analytic geometry.

🔑 Key Principles

• 📐 Linear Equation Form: A linear equation typically takes the form $ax + by = c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.
• 📍 Point Representation: A point in a two-dimensional plane is represented as an ordered pair $(x, y)$.
• ✅ Substitution: To check if a point satisfies a system, substitute the $x$ and $y$ values of the point into each equation in the system.
• ⚖️ Verification: If the substitution results in a true statement for every equation in the system, the point satisfies the system. Otherwise, it does not.

🌍 Real-world Examples

Example 1: Consider the system of equations:

$\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases}$

Let's check if the point $(2, 3)$ satisfies this system.

• ➕ For the first equation: $2 + 3 = 5$, which is true.
• ➖ For the second equation: $2(2) - 3 = 4 - 3 = 1$, which is true.

Since $(2, 3)$ satisfies both equations, it satisfies the system.

Example 2: Consider the system of equations:

$\begin{cases} x - y = 2 \\ x + y = 4 \end{cases}$

Let's check if the point $(1, 1)$ satisfies this system.

• ➖ For the first equation: $1 - 1 = 0 \neq 2$, which is false.
• ➕ For the second equation: $1 + 1 = 2 \neq 4$, which is false.

Since $(1, 1)$ does not satisfy either equation, it does not satisfy the system.

📊 Tabular Example

💡 Conclusion

Understanding when a point satisfies a system of linear equations is fundamental in algebra and has broad applications in various fields. By substituting the coordinates of a point into the equations and verifying their truth, we can determine whether the point is a solution to the system. This concept is crucial for solving systems of equations and understanding their graphical representations.