Bot GPT 4-o mini To find the equation of the line in slope-intercept form \(y = mx + b\) using the points (5, -5) and (1, 5), we will first calculate the slope \(m\) and then use one of the points to find the y-intercept \(b\).
- Calculate the slope \(m\) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (5, -5) as \((x_1, y_1)\) and (1, 5) as \((x_2, y_2)\):
\[ m = \frac{5 - (-5)}{1 - 5} = \frac{5 + 5}{1 - 5} = \frac{10}{-4} = -\frac{5}{2} \]
- Use the slope and one of the points to find \(b\). Let's use the point (5, -5):
\[ y = mx + b \] \[ -5 = -\frac{5}{2}(5) + b \] \[ -5 = -\frac{25}{2} + b \]
Now, add \(\frac{25}{2}\) to both sides to solve for \(b\):
\[ b = -5 + \frac{25}{2} \] Convert -5 to a fraction with a denominator of 2:
\[ -5 = -\frac{10}{2} \] So,
\[ b = -\frac{10}{2} + \frac{25}{2} = \frac{15}{2} \]
- Write the equation in slope-intercept form:
Now that we have \(m = -\frac{5}{2}\) and \(b = \frac{15}{2}\), we can write the equation:
\[ y = -\frac{5}{2}x + \frac{15}{2} \]
Thus, the equation of the line in slope-intercept form is:
\[ y = -\frac{5}{2}x + \frac{15}{2} \]
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