91Ó°ÊÓ

When discussing analytic geometry, understanding if a point lies on a graph is crucial. Imagine you have a graph on a coordinate plane, similar to a treasure map, and each point is like an 'X' marking a location. To find out if a specific point, for example, \((1, -2)\), is part of the path of a treasure hunt marked by an equation, you need to test it. The coordinates \((x, y)\) represent locations. If these coordinates satisfy the equation, the point is indeed on the graph. Otherwise, it's like having a wrong direction in your map. By substituting these coordinates into the equation and checking if the equation holds true, you can determine the correctness of the point's position on the graph.

Linear equations in two variables, like ours, can be thought of as a straight path or line on a graph. The general form is usually \(ax + by = c\). In our situation, the given equation is \(3x + y = 12\). Linear equations are called 'linear' because when they are plotted on a coordinate plane, they form a straight line. This line represents all the possible solutions of the equation where every point on the line makes the equation true. Thus, any point that satisfies the equation, places nicely along this line, much like an accurate step along a path.

Coordinate substitution is a method used to verify if a point lies on a graph. Think of it as entering variables' values to solve a mathematical puzzle. In our problem, we substitute the given point's coordinates \(x = 1\) and \(y = -2\) into the equation \(3x + y = 12\). This involves replacing \(x\) and \(y\) in the equation with these values. After substitution, we evaluate the equation. If the left-hand side equals the right-hand side, then the point lies on the equation's line. If not, as in \(1 = 12\), then the point does not lie on the graph. This simple method helps in determining immediate accuracy of the point’s location on the line described by the equation.