Question

Graph the line that passes through the points. Write its equation in slope- intercept form. (Review \(5.3 \text { for } 5.7)\) $$(5,-8),(7,-3)$$

Short answer

The equation of the line that passes through the points \((5,-8)\) and \((7,-3)\) is \(y = \frac{5}{2}x - \frac{41}{2}\)

Step-by-step solution

Find the slope (m)

The slope of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let's use the points \((5,-8)\) and \((7,-3)\) and replace the variables in the formula with the corresponding values from these points. The calculation is as follows: \(m = \frac{-3 - (-8)}{7 - 5} = \frac{5}{2}\). So, the slope of the line is \(5/2\)

Find the y-intercept (b)

To find \(b\), we'll use the slope-intercept equation \(y = mx + b\) and rearrange it to find \(b = y - mx\). We know that \(m = \frac{5}{2}\) and we can take one of the given points, say \((5,-8)\), for the values of \(x\) and \(y\). So, \(b = -8 - (5/2)*5 = -8 - \frac{25}{2} = -\frac{41}{2}\). So, the y-intercept of the line is \(-\frac{41}{2}\)

Write the equation of the line

Now we introduce the values of \(m\) and \(b\) that we found previously into the slope-intercept form to obtain the equation of the line. The slope-intercept form of the equation of a line is \(y = mx + b\). By substituting, our equation becomes \(y = \frac{5}{2}x - \frac{41}{2}\)

Graph the line

To graph this line, start by plotting the y-intercept \(-\frac{41}{2}\) on the y-axis. Then, from there, use the slope \(\frac{5}{2}\) to find the next point. The slope is the ratio of 'rise over run', so from the y-intercept, count 5 units up (rise is positive 5) and 2 units to the right (run is positive 2). Plot this new point. Draw a straight line through these two points, extending it beyond them in both directions.