Question

Find an equation of the line satisfying the conditions. Write your answer in the slope-intercept form. Passes through \((-1,-2)\) and \((3,-4)\)

Short answer

The equation of the line passing through the points \((-1, -2)\) and \((3, -4)\) in slope-intercept form is \(y = -\frac{1}{2}x - \frac{5}{2}\).

Step-by-step solution

Find the slope

First, we need to find the slope of the line passing through the two given points. The slope formula is given by: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] Using the given points \((-1, -2)\) and \((3, -4)\), we can substitute the coordinates into the formula: \[m = \frac{-4 - (-2)}{3 - (-1)}\]

Simplify the slope

Now, simplify the expression for the slope: \[m = \frac{-4 + 2}{3 + 1}\] \[m = \frac{-2}{4}\] \[m = -\frac{1}{2}\] So the slope (m) of the line is -1/2.

Use the point-slope form

Next, we will use the point-slope form to find the equation of the line. The point-slope form is given by: \[y - y_1 = m(x - x_1)\] We can use either of the given points, so we will use \((-1, -2)\) and our calculated slope: \[y - (-2) = -\frac{1}{2}(x - (-1))\]

Write the equation in slope-intercept form

Now, let's rewrite the equation in the slope-intercept form, which is \(y = mx + b\). 1. Begin by simplifying the equation: \[y + 2 = -\frac{1}{2}(x+1)\] 2. Distribute the slope to both terms inside the parentheses: \[y + 2 = -\frac{1}{2}x - \frac{1}{2}\] 3. Subtract 2 from both sides of the equation to isolate y: \[y = -\frac{1}{2}x - \frac{1}{2} - 2\] 4. Combine the constants into a single term: \[y = -\frac{1}{2}x - \frac{5}{2}\] So, the equation of the line is \(y = -\frac{1}{2}x - \frac{5}{2}\), written in slope-intercept form.