Describe two ways to determine whether or not a given point is a solution of a line. Which way is more accurate? Why?

The only practical and fast way is to substitute the coordinates of the point into the equation.
If the equation is "true", then it is a solution.

Any other method is inefficient and impractical

Determining whether a given point is a solution of a line can be done through two methods. The first and most accurate way is to substitute the coordinates of the point into the equation representing the line. The second method, which is less accurate, involves plotting the point on a graph and visually examining whether it lies on the line.

To substitute the coordinates of the point into the equation, you'll need the equation of the line in either slope-intercept form (y = mx + b) or standard form (Ax + By = C). Let's say you have the equation of the line in slope-intercept form, y = mx + b. To determine if a point (x, y) is a solution of the line, you substitute the x-coordinate of the point into the equation in place of x, and the y-coordinate in place of y. If the resulting equation is true (both sides of the equation are equal), then the point is a solution of the line. If it is not true, then the point is not a solution.

For example, let's say you have the equation of a line as y = 2x + 3 and you want to determine if the point (1, 5) is a solution. Substituting the x-coordinate of 1 and the y-coordinate of 5 into the equation, you get 5 = 2(1) + 3. Simplifying it, 5 = 2 + 3 which is indeed true. Therefore, the point (1, 5) is a solution of the line.

The second method involves plotting the point on a graph and visually examining whether it lies on the line. You can draw the line using its equation and then plot the given point on the graph. If the point lies on the line, it is a solution. However, this method is less accurate because visual inspection can be subjective and prone to human error. It may not be precise, especially if the graph is not accurately drawn or if the point lies very close to the line.

Substituting the coordinates of the point into the equation is the more accurate method because it provides an exact and definite answer. It follows the precise mathematical relationship defined by the equation. On the other hand, visually inspecting the graph may lead to approximations and subjective judgments. Therefore, when accuracy is crucial, it is recommended to rely on the method of substituting coordinates into the equation.