To find the equation of a line in slope-intercept form [tex]\( y = mx + b \)[/tex] from a given table of points, we need to determine the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex].
1. Determine the Slope (m):
- Select any two points from the table. Let's use the points [tex]\((3, -3)\)[/tex] and [tex]\((0, -2)\)[/tex].
- The formula for the slope [tex]\( m \)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- Substituting the points [tex]\((3, -3)\)[/tex] and [tex]\((0, -2)\)[/tex]:
[tex]\[
m = \frac{-2 - (-3)}{0 - 3} = \frac{-2 + 3}{0 - 3} = \frac{1}{-3} = -\frac{1}{3}
\][/tex]
2. Confirming the Slope with Another Pair of Points:
- Let's use another pair, such as [tex]\((0, -2)\)[/tex] and [tex]\((-3, -1)\)[/tex]:
[tex]\[
m = \frac{-1 - (-2)}{-3 - 0} = \frac{-1 + 2}{-3} = \frac{1}{-3} = -\frac{1}{3}
\][/tex]
- The slope remains consistent, confirming our calculation. Thus, [tex]\( m = -\frac{1}{3} \)[/tex].
3. Determine the Y-Intercept (b):
- Use the slope-intercept form: [tex]\( y = mx + b \)[/tex].
- Plug in one of the points, for instance, [tex]\((0, -2)\)[/tex]:
[tex]\[
-2 = -\frac{1}{3}(0) + b
\][/tex]
[tex]\[
-2 = b
\][/tex]
- Thus, [tex]\( b = -2 \)[/tex].
4. Formulate the Equation:
- Substitute the slope [tex]\( m = -\frac{1}{3} \)[/tex] and y-intercept [tex]\( b = -2 \)[/tex] back into the slope-intercept form:
[tex]\[
y = -\frac{1}{3}x - 2
\][/tex]
Therefore, the equation for the line in slope-intercept form is:
[tex]\[
\boxed{y = -\frac{1}{3}x - 2}
\][/tex]
From the given options, the correct choice is:
[tex]\[
y = -\frac{1}{3}x - 2
\][/tex]
