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Coordinate geometry, also known as analytic geometry, is all about understanding and representing geometric shapes using algebraic equations. It deals with the coordinate plane, where each point is identified by two numbers - the x-coordinate and the y-coordinate. These coordinates are like addresses that tell us exactly where a point is located on the grid.

In coordinate geometry, we mainly work with lines, curves, and figures plotted in the coordinate plane. Every line can be described by a linear equation, which brings us to another important concept: the equation of a line. By substituting different values of x and y, you can determine whether specific points lie on the line represented by the equation. This allows you to clearly visualize and understand relationships between geometric shapes and algebra.

The substitution method is a key technique for solving equations and verifying the solutions. When checking if a point lies on a line, we substitute the x and y values from the coordinate point into the given equation. This helps us see if both sides of the equation match.

For example, if we have the equation \(2x + 5y = 6\) and want to check if the point (-2, 2) lies on it, we substitute \(x = -2\) and \(y = 2\) into the equation. We calculate \(2(-2) + 5(2)\) and see if it equals 6. If the results on both sides of the equation are equal, then the point truly lies on the graph of the line. The substitution method is like putting puzzle pieces into place to see if they fit.

The equation of a line is a way to express every point on that line mathematically. Linear equations usually come in the form \(y = mx + c\), known as the slope-intercept form, or in standard form \(Ax + By = C\), like the equation \(2x + 5y = 6\) in our example.

These equations are all about forming a relationship between x and y coordinates. For each x-value, there corresponds a y-value that satisfies the equation. When you change x, y changes based on the equation, thereby tracing out the line described by the equation. Understanding these equations is like understanding the language lines use to speak about their position and direction.

Solving equations means finding values for the variables that make the equation true. In our linear equation example, \(2x + 5y = 6\), we need specific x and y values so that the left side equals the right side.

To solve for x or y, we can rearrange the equation to isolate one of the variables. This often involves performing operations like adding, subtracting, multiplying, or dividing, just like normal arithmetic. When checking specific points, like (-2, 2) or (3, 0), against the equation, we substitute the point values back into the equation to see if it holds true. This process is like working through a mystery, finding the right numbers to balance the equation.