Question

Use a graph to determine whether the given three points seem to lie on the same line. If they do, prove algebraically that they lie on the same line and write an equation of the line. $$ (-3,-1),(0,1),(12,9) $$

Short answer

By visual inspection and algebraic confirmation, points (-3,-1), (0,1), and (12,9) lie on the line given by the equation \(y = \frac{5}{4}x + 1\).

Step-by-step solution

Visual Inspection

Plot the three points (-3,-1), (0,1), and (12,9) on a graph. Based on the graph, you may be able to visually confirm if the points lie on the same line.

Calculate Slopes

Calculate the slope between the pairwise combinations of points: (-3,-1), (0,1) and (0,1), (12,9). Use the formula to calculate slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\). If all the slopes are equal, it suggests these points are on the same line.

Algebraic Proof

Use the third point (12,9) to confirm if it lies on the line defined by the other two points. Substitute coordinates of the third point in the equation and confirm the equality.

Equation of the Line

If the three points are confirmed to be on the same line, the equation of the line can now be written. For this, use the slope calculated in step 2 and coordinates of any point. The equation of the line in slope-intercept form is \(y = mx + b\), where m is the slope and b is the y-intercept.