Problem 34 - 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 \((-2,-5)\) and \((6,-5)\)

Step-by-Step Solution

Answer

The equations of the line in both point-slope form and slope-intercept form is \(y = -5\), which is a horizontal line at -5 on the y-axis.

Step 1: Calculate the Slope of the Line

The slope of a line (m) passing through two known points \((x_1,y_1)\) and \((x_2,y_2)\) is calculated by the formula \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \). Applying this formula to the given coordinates, we get \( m = \frac{{-5 - (-5)}}{{6 - (-2)}} = 0 \). From here we can see that the slope is 0, which means the line is horizontal.

Step 2: Write the Equation in Point Slope Form

The point-slope form of a line's equation is given by \(y - y_1 = m(x - x_1)\). Here, \(m\) is the slope and \(x_1, y_1\) are the coordinates of a point on the line. We can use one of our given points and the calculated slope for this form, i.e., \(y - (-5) = 0(x - (-2))\), which simplifies to \(y = -5\).

Step 3: Write the Equation in Slope Intercept Form

The slope-intercept form of a line's equation is given as \(y = mx + c\), where m is the slope and c is the y-intercept. In this case, as the line is horizontal and the y coordinate remains constant while the x coordinate changes, the y-intercept is -5. So the equation becomes \(y = 0x - 5\) which simplifies to \(y = -5\).