91Ó°ÊÓ

Coordinate geometry, or analytic geometry, is a branch of mathematics that uses algebraic equations to represent geometric figures. It involves understanding how points, lines, and other shapes behave on a coordinate plane. A coordinate plane, also known as a Cartesian plane, consists of two perpendicular lines: the horizontal axis (x-axis) and the vertical axis (y-axis). Every point in this plane can be represented by an ordered pair \(x, y\).

A linear equation — like \(y = 4x - 9\) from our exercise — describes a straight line in coordinate geometry. Each point on this line is a solution to the equation, meaning if you substitute its \(x\) and \(y\) values into the equation, they satisfy it and maintain its balance.

• No two points with different \(x\) values will correspond to the same \(y\) value on the line, making the relationship unique and linear.
• Graphically, you could plot these points to visualize if they lie on the line derived from the equation.

The substitution method is a way of finding out whether a given point lies on the line represented by a linear equation. It involves replacing the variables in the equation with the coordinates of the point.

In our exercise, for example, each point was tested against the equation \(y = 4x - 9\). Here's how you do it:

• Take a point, say \((1, -2)\).
• Substitute \(x = 1\) in the equation: \(y = 4(1) - 9\).
• This gives \(y = 4 - 9 = -5\).
• Compare this \(y\) value to the \(y\) in the point (which is -2). They do not match, so it's not on the line.

Repeat the process for other points to determine if they are part of the line or not. This method helps verify solutions by checking their consistency with the equation.

Solution verification in mathematics, particularly in linear equations, is about confirming whether a particular point satisfies the given equation. It's a crucial step that ensures the accuracy of results.

Once you have used the substitution method, verify each solution by:

• Checking if the derived \(y\) values exactly match the \(y\) coordinates of the points you are testing.
• If they match, the point lies on the line represented by the equation; otherwise, it does not.
• This process removes errors by ensuring logic and calculations align perfectly.

In our example, point \(8, 23\) was verified as a solution because substituting \(x = 8\) resulted in \(y = 23\), which matched the \(y\) coordinate of the point.