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When working with linear equations, coordinates play a crucial role. Coordinates are a pair of numbers that determine the precise location of a point on a plane. In a two-dimensional space, these coordinates are written as \(x, y\), representing the horizontal and vertical positions, respectively.

In the given exercise, we have the point (-1,0). Here, -1 is the x-coordinate, which determines how far left or right the point is from the origin, and 0 is the y-coordinate, which shows how far up or down the point is from the origin.

• The x-coordinate tells you how much to move horizontally.
• The y-coordinate tells you how much to move vertically.

Understanding the coordinates is the first essential step in analyzing whether the point lies on a specific line, as it allows you to input accurate values into the linear equation.

The method of substitution is an algebraic technique used to solve linear equations or determine if specific points lie on a line. It involves replacing variables with given numbers, in this case, coordinates.

For the exercise, we substitute \(x = -1\) and \(y = 0\) into the line equation \(y = -2x + 5\). This substitution results in the equation \(0 = -2(-1) + 5\).

• Replace the x in the equation with the x-coordinate of the point.
• Replace the y in the equation with the y-coordinate of the point.

This step is crucial because it transforms the general equation into a specific one that can be evaluated to see if the input coordinates satisfy the equation.

Equation validation is the step where you determine whether both sides of the equation are equal after substitution. This process confirms if a point lies on the given line.

After substituting the coordinates into the line equation, we simplify it: \(0 = 2 + 5\). At this point, you'll notice that the equation simplifies to \(0 = 7\), which is clearly incorrect.

• If the sides are equal, the point satisfies the line equation, indicating it lies on the line.
• If the sides are not equal, like in this case, then the point does not lie on the line.

This validation step is critical because it conclusively tells us whether or not the point is part of the line described by the equation. In this exercise, since \(0 eq 7\), the point \((-1,0)\) does not lie on the line \(y = -2x + 5\).