A linear equation in two variables is an equation that can be written in the form:
Ax + By = C
Where:
- x and y are the variables.
- A, B, and C are constants, and A and B are not both equal to zero.
In class 10th mathematics, you typically learn how to work with linear equations in two variables. Here are some key concepts and techniques related to linear equations in two variables:
- Graphical Representation: You can graphically represent a linear equation by plotting its points on a coordinate plane. The graph of a linear equation is a straight line.
- Solving Simultaneous Linear Equations: One of the fundamental applications of linear equations in two variables is solving systems of simultaneous linear equations. You can do this using various methods, including:
- Substitution Method: Solve one equation for one variable and substitute it into the other equation.
- Elimination Method: Add or subtract equations to eliminate one variable and solve for the other.
- Intercepts: The x-intercept is the point where the line crosses the x-axis, and the y-coordinate is zero. The y-intercept is the point where the line crosses the y-axis, and the x-coordinate is zero.
- Slope and Slope-Intercept Form: The slope of a line is the ratio of the change in y to the change in x between two points on the line. The slope-intercept form of a linear equation is:y = mx + bWhere:
- m is the slope of the line.
- b is the y-intercept, the value of y when x is zero.
- Parallel and Perpendicular Lines: Lines with the same slope are parallel, while lines with slopes that are negative reciprocals of each other are perpendicular.
- Word Problems: Linear equations in two variables are used to solve real-world problems involving relationships between two variables. You’ll encounter word problems where you’ll need to set up and solve linear equations.
Here’s a simple example of solving a linear equation in two variables:
Example: Solve the system of equations:
- 2x + 3y = 7
- 4x – 2y = 6
You can use the elimination method or substitution method to solve this system of equations and find the values of x and y.
Linear equations in two variables are an essential topic in mathematics and serve as a foundation for more advanced concepts in algebra and other fields of mathematics.