Linear Equations in Two Variables
This section delves into the concept of linear equations in two variables. A linear equation can be generally expressed in the form ax + by + c = 0, where a and b are not both zero. It emphasizes the relationship between the variables and introduces the idea of solutions, which are ordered pairs (x, y). Unlike linear equations in one variable, linear equations in two variables can generate infinitely many solutions, as demonstrated through various examples. Additionally, the section provides exercises aimed at transforming equations into the standard form and encourages problem-solving as a method to understand the relationships between variables in real-life contexts.
Example 2 : Write each of the following as an equation in two variables:
(i) \( x = 7 \)
(ii) \( y = 3 \)
(iii) \( 5x = 15 \)
(iv) \( 4y = 20 \)
Solution :
(i) \( x = 7 \) can be written as \( 1 \cdot x + 0 \cdot y = 7 \), \( 0 \cdot x + 1 \cdot y + 7 = 0 \).
(ii) \( y = 3 \) can be written as \( 0 \cdot x + 1 \cdot y - 3 = 0 \).
(iii) \( 5x = 15 \) can be written as \( 5 \cdot x - 15 = 0 \).
(iv) \( 4y = 20 \) can be written as \( 0 \cdot x + 4 \cdot y - 20 = 0 \).