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. $$ (-2,-1),(3,2),(7,5) $$

Short answer

The three points (-2,-1), (3,2), and (7,5) do lie on the same line. The constant slope between the points is \(\frac{1}{1}\). The equation of the line on which the points lie can be written as \(y - (-1) = 1 *(x - (-2))\), simplified to \(y = x + 1\).

Step-by-step solution

Graphical Determination

Initially, plot the three points (-2,-1), (3,2), and (7,5) on a graph. If all the three points lie on the same line, proceed to step 2.

Compute Slope

Next, calculate the slope between the points. The formula for slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In this case, calculate the slope between point 1 and point 2, and point 2 and point 3.

Compare Slopes

If the slopes computed in step 2 are equal, that's an indication that the three points lie on the same line. This is because a straight line has a constant slope.

Equation of the Line

Once it is confirmed that the points are collinear i.e., they lie on the same line, then the equation of the line that they lie on can be found. Use the slope formula \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is any of the three points. Substituting one set of coordinates and the slope will give the equation of the line