Linear Equation in Two Variables:
A linear equation in two variables is an equation where two variables are involved, and the degree of each variable is one.
Standard Form: It can be written in the form \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are real numbers.
- \(x\) and \(y\) are variables.
- \(a\) and \(b\) are not both zero.
- The variables are arranged so that they are not multiplied by each other.
Example:
\(x + y = 143\) , \(20s + 10t = 225\).
Solution of a Linear Equation:
A solution of an equation is a number (or a pair of numbers for a two-variable equation) substituted for the unknown variable(s) that makes the equality in the equation true.
- In a two-variable equation, the solution is an ordered pair \((x, y)\).
- A single linear equation in two variables can have many solutions.
Method to Find Solutions:
Method I (Substitution): Substitute values for \(x\) and \(y\) into the LHS and check if it equals the RHS.
Method II (Intercepts): Substitute \(x = 0\) to find the \(y\)-intercept, and substitute \(y = 0\) to find the \(x\)-intercept. The resulting pairs, e.g., \((0, 10)\) and \((20, 0)\) for \(x + 2y = 20\), are solutions.
Properties of Equations:
The solution of a linear equation is not affected when:
- The same number is added to (or subtracted from) both sides of the equation.
- You multiply or divide both sides of the equation by the same non-zero number.
Geometrical Representation (Graph):
The geometrical representation of a degree one polynomial equation (\(ax + by + c = 0\)) is a straight line. This is why it is called a linear equation.
Relationship between Points and Solutions:
- Every point whose coordinates satisfy the equation lies on the line of the graph.
- Every point \((a, b)\) on the line gives a solution \(x = a, y = b\) of the given equation.
- Any point that does not lie on the line is not a solution to the given equation.
Important!
Equation \(y = kx\): The graph of an equation in the form \(y = kx\) is a line that always passes through the origin.
Equations of Lines Parallel to the Axes:
Line parallel to the \(x\)-axis:
The equation for such a line is \(y = b\), where \(b\) is the constant value of the \(y\)-coordinate of any point on the line.
Important!
Every point on the \(x\)-axis is of the form \((x, 0)\).
Line parallel to the \(y\)-axis:
The equation for such a line is \(x = a\), where \(a\) is the constant value of the \(x\)-coordinate of any point on the line.
Important!
Every point on the \(y\)-axis is of the form \((0, y)\).