1. Introduction
A linear equation in two variables is an equation that contains two variables, usually represented by \(x\) and \(y\), and each variable has power 1. These equations represent straight lines when plotted on a graph.
Examples include:
- \(2x + 3y = 7\)
- \(x - y = 4\)
- \(3a + 2b = 12\)
Unlike linear equations in one variable, these equations have infinitely many solutions.
2. Standard Form of a Two-Variable Linear Equation
The standard form of a linear equation in two variables is:
\(ax + by + c = 0\)
Here:
- \(a, b, c\) are real numbers
- \(a\) and \(b\) cannot both be zero
Any equation that can be converted to this form is a linear equation in two variables.
2.1. Examples
- \(2x + 5y - 3 = 0\)
- \(7x - y + 9 = 0\)
- \(4a + 3b = 12\)
3. Solution of a Linear Equation in Two Variables
A solution of a linear equation in two variables is an ordered pair \((x, y)\) that satisfies the equation.
Because there are two variables, such equations have infinitely many solutions. Each solution represents a point on the straight line formed by the equation.
3.1. What is an Ordered Pair?
An ordered pair is written as \((x, y)\). It shows the value of each variable.
Example:
- \((2, 3)\) means \(x = 2\) and \(y = 3\)
- \((-1, 4)\) means \(x = -1\) and \(y = 4\)
3.2. Checking if an Ordered Pair is a Solution
To check if \((x, y)\) is a solution, substitute the values into the equation.
Example:
Check if \((2, 1)\) satisfies \(2x + 3y = 7\).
- \(2(2) + 3(1) = 4 + 3 = 7\)
- Yes, it is a solution.
3.2.1. More Examples
- \((1, 2)\) satisfies \(x + 2y = 5\)
- \((3, 0)\) satisfies \(3x - y = 9\)
- \((0, 4)\) does not satisfy \(2x + y = 3\)
4. Writing Linear Equations from Statements
Many real-life situations can be written as linear equations involving two variables.
4.1. Examples
- If the sum of two numbers is 10: \(x + y = 10\)
- The cost of 2 pens and 3 pencils is 25: \(2x + 3y = 25\)
- A rectangle has perimeter 20: \(2l + 2b = 20\)
5. Value Table Method
We can find multiple solutions of a linear equation by choosing a value for one variable and calculating the other. This is called the value table method.
5.1. Example
For the equation \(2x + y = 7\):
| \(x\) | \(y\) |
|---|---|
| 0 | 7 |
| 1 | 5 |
| 2 | 3 |
| 3 | 1 |
6. Graphical Understanding
Each solution of a linear equation in two variables corresponds to a point on its graph (a straight line). Plotting several solutions gives a straight line.
Example:
Equation: \(x + y = 4\)
- Solutions include \((0,4), (1,3), (2,2), (3,1), (4,0)\)
Plotting these gives a straight line.
7. Quick Practice
Try these questions:
- Find whether \((3, 2)\) satisfies \(2x + y = 8\).
- Write an equation whose solution is \((2, 5)\).
- Make a value table for \(x - 2y = 4\).
- Check two solutions of \(3x + y = 10\).
8. Summary
- A linear equation in two variables has the form \(ax + by + c = 0\).
- Solutions are written as ordered pairs \((x, y)\).
- Such equations have infinitely many solutions.
- Each solution corresponds to a point on a straight-line graph.
- Value tables help generate multiple solutions easily.