Problem 25 - 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 \((1,2)\) and \((5,10)\)

Step-by-Step Solution

Answer

The point-slope form of the equation of the line is \(y - 2 = 2(x - 1)\) which simplifies to \(y = 2x\). The slope-intercept form of the equation of the line is also \(y = 2x\).

Step 1: Determine the Slope

The slope of a line passing through the points \((x_1,y_1)\) and \((x_2,y_2)\) is given by the formula \(m = (y_2 - y_1)/(x_2 - x_1)\). Here, \(x_1 = 1\), \(y_1 = 2\), \(x_2 = 5\), and \(y_2 = 10\). So, \(m = (10 - 2)/(5 - 1) = 2.\)

Step 2: Write the Point-Slope Form

The point-slope form of the equation is given by \(y - y_1 = m(x - x_1)\). Substituting \(m = 2\), \(x_1 = 1\), and \(y_1 = 2\) gives \(y - 2 = 2(x - 1)\). Simplifying, we get \(y = 2x\).

Step 3: Write the Slope-Intercept Form

The slope-intercept form is \(y = mx + c\). Substituting \(m = 2\) and rearranging the point-slope equation, we find that the y-intercept \(c = 0\). Thus, the slope-intercept form of the equation is \(y = 2x\).