91Ó°ÊÓ

Coordinate geometry is a branch of mathematics that provides a link between algebra and geometry. It utilizes a pair of axes known as the Cartesian coordinates to represent points in a plane. In simple terms, the position of any point in the plane can be described by two numbers, usually represented as \( (x, y) \), where \( x \) corresponds to the horizontal distance from a reference point, and \( y \) corresponds to the vertical distance.

In coordinate geometry, various geometric shapes can be analyzed, such as lines, circles, and polygons, by using algebraic equations. For instance, the equation of a line in the plane can be given in the form \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept. To determine if a specific point lies on a line, you simply substitute the point's coordinates into the line's equation and check for equality.

Linear equations are foundational in algebra and represent straight lines when graphed on the coordinate plane. They are called 'linear' because the power of both \( x \) and \( y \) is 1. The general form of a linear equation is \( Ax + By = C \), where \( A \) and \( B \) are coefficients, \( x \) and \( y \) are variables, and \( C \) is a constant.

These equations can often be solved using various methods such as graphing, substitution, elimination, or using matrices. Solving a linear equation like \( 2x - y = 4 \) involves finding all pairs of \( (x, y) \) that make the equation true. If we have additional information, like a point through which the line passes, we can plug the point's coordinates into the equation to verify its validity on the line.

The substitution method is a technique to solve systems of equations. This approach is useful when one of the equations in the system can be easily solved for one variable in terms of the others. Then, this expression is 'substituted' into the other equation(s), reducing the number of variables and equations.

In the context of a single linear equation, substitution involves replacing the variables with specific values to evaluate the equation's validity. For example, given a point \( (0,0) \) and an equation \( 2x - y = 4 \) you would substitute the x-coordinate ('0') and the y-coordinate ('0') into the equation to determine if the point satisfies the equation. With simple substitution, we see that \( 2 \times 0 - 0 \) does not equal 4, which indicates the point \( (0,0) \) is not on the line defined by the equation.