Problem 27 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-3,0)\) and \((0,3)\)

Step-by-Step Solution

Answer

The equation of the line passing through the points \((-3,0)\) and \((0,3)\), in point-slope form is \(y = x + 3\), and in slope-intercept form is \(y = x + 3\).

Step 1: Finding the slope

The formula for calculating the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). By substituting the given points into this formula, you get \(m = \frac{3 - 0}{0 - -3} = 1\). The slope of the line is thus 1.

Step 2: Writing the point-slope form of the line

The point-slope form of a line is \(y - y_1 = m(x - x_1)\). Here, \(m\) is the slope and \((x_1, y_1)\) are the coordinates of a point on the line. By using the calculated slope and one of the given points, for example \((-3,0)\), we get: \(y - 0 = 1(x - -3)\), which simplifies to \(y = x + 3\).

Step 3: Writing the slope-intercept form of the line

The point-slope form \(y = x + 3\) is actually already in slope-intercept form. The slope-intercept form is \(y = mx + b\), where m is the slope and b is the y-intercept. Thus, comparing the two forms, we can see that the y-intercept is 3.