91Ó°ÊÓ

Linear equations form the foundation of algebra and define straight lines in coordinate geometry. They are algebraic equations in which each term is either a constant or the product of a constant and a single variable. A standard form of a linear equation in two variables, like x and y, is presented as y = mx + b, where m is the slope of the line and b is the y-intercept.

The slope is a measure of how steep the line is, indicating the rate at which y changes with respect to x. On the other hand, the y-intercept gives the point through which the line crosses the y-axis, that is, the value of y when x equals zero. In the context of the mentioned exercise, the equation y = -2x + 5 describes a line that slopes downwards as it moves from left to right, with a slope of -2 and crossing the y-axis at (0, 5).

The substitution method is a technique for solving systems of equations or checking whether a point lies on a certain line. This strategy involves substituting the values from one equation into another to solve for the variables. The method is straightforward and highly effective when dealing with linear equations.

In our exercise, the substitution method is used by inserting the x-coordinate (in this case, 0) and the y-coordinate (in this case, 3) from the point (0, 3) into the equation y = -2x + 5. If after the substitution the resulting statement is true, then the point lies on the line. If not, as seen in the solution where the equation 3 = 5 does not hold, the point does not lie on the line.

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This field of mathematics allows for the graphical representation of geometric figures and the analysis of their properties. The Cartesian coordinate system is most commonly used, which consists of two axes, the x-axis and the y-axis, that intersect at a point called the origin.

Points on the plane are identified by their coordinates, expressed as a pair (x, y). In the exercise, the task is to verify if the point (0, 3) is located on the line represented by the equation y = -2x + 5. Coordinate geometry is crucial in this case because it allows us to graphically represent the line and points, providing a visual tool to understand the position of points relative to the line.