To find the equation of the line that passes through the points [tex]\((-3, 11)\)[/tex] and [tex]\((3, -1)\)[/tex] in slope-intercept form [tex]\(y = mx + b\)[/tex], we need to determine both the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex].
Step 1: Calculate the slope [tex]\(m\)[/tex]
The formula to calculate the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Given the points:
- [tex]\((x_1, y_1) = (-3, 11)\)[/tex]
- [tex]\((x_2, y_2) = (3, -1)\)[/tex]
Substitute these values into the slope formula:
[tex]\[ m = \frac{-1 - 11}{3 - (-3)} \][/tex]
[tex]\[ m = \frac{-12}{6} \][/tex]
[tex]\[ m = -2 \][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(-2\)[/tex].
Step 2: Find the y-intercept [tex]\(b\)[/tex]
The slope-intercept form of the equation [tex]\(y = mx + b\)[/tex] can be rearranged to find the y-intercept [tex]\(b\)[/tex]:
[tex]\[ b = y - mx \][/tex]
Use one of the given points to find [tex]\(b\)[/tex]. We'll use the point [tex]\((-3, 11)\)[/tex].
Substitute [tex]\(m = -2\)[/tex], [tex]\(x = -3\)[/tex], and [tex]\(y = 11\)[/tex]:
[tex]\[ b = 11 - (-2)(-3) \][/tex]
[tex]\[ b = 11 - 6 \][/tex]
[tex]\[ b = 5 \][/tex]
So, the y-intercept [tex]\(b\)[/tex] is [tex]\(5\)[/tex].
Step 3: Form the equation of the line
Now that we have both the slope [tex]\(m = -2\)[/tex] and the y-intercept [tex]\(b = 5\)[/tex], we can write the equation of the line in slope-intercept form:
[tex]\[ y = -2x + 5 \][/tex]
Therefore, the equation of the line that passes through the points [tex]\((-3, 11)\)[/tex] and [tex]\((3, -1)\)[/tex] is:
[tex]\[ y = -2x + 5 \][/tex]
