Problem 19 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-1,\) passing through \(\left(-\frac{1}{2},-2\right)\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 2 = -x - 1/2\). In slope-intercept form, it is \(y = -x - 5/2\).

Step 1: Identify slope and given point

The slope of the line is given as -1 and the line passes through the point (-1/2, -2). These two pieces of information will allow us to write the equation in both forms.

Step 2: Write the Point-Slope form

Using the point-slope form \(y-y_1=m(x-x_1)\), substitute \(m=-1, x_1= -1/2,\) and \(y_1=-2\). After substituting these values, the equation becomes \(y - (-2) = -1(x -(-1/2))\) which simplifies to \(y + 2 = -1(x +1/2)\). You can further simplify this to get \(y + 2 = -x - 1/2\). This is the point-slope form of the line.

Step 3: Write the Slope-Intercept form

For the slope-intercept form, solve the equation from Step 2 for \(y\). This will give \(y = -x - 1/2 - 2\) which simplifies to \(y = -x - 5/2\). This is the slope-intercept form of the line.