To determine the slope-intercept form of the equation for the line passing through the points given in the table, we need to find the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex].
The points given are:
[tex]\[
(-5, -\frac{11}{5}), (-3, -\frac{7}{5}), (1, \frac{1}{5}), (3, 1), (5, \frac{9}{5})
\][/tex]
### Step 1: Find the Slope ([tex]\(m\)[/tex])
The formula for 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]
Choose the first two points [tex]\((-5, -\frac{11}{5})\)[/tex] and [tex]\((-3, -\frac{7}{5})\)[/tex]:
[tex]\[
m = \frac{-\frac{7}{5} - (-\frac{11}{5})}{-3 - (-5)}
\][/tex]
Simplify the numerator:
[tex]\[
-\frac{7}{5} - (-\frac{11}{5}) = -\frac{7}{5} + \frac{11}{5} = \frac{4}{5}
\][/tex]
Simplify the denominator:
[tex]\[
-3 - (-5) = -3 + 5 = 2
\][/tex]
Now put it all together:
[tex]\[
m = \frac{\frac{4}{5}}{2} = \frac{4}{5} \times \frac{1}{2} = \frac{4}{10} = \frac{2}{5}
\][/tex]
This results in a slope [tex]\(m = \frac{2}{5}\)[/tex].
### Step 2: Find the y-intercept ([tex]\(b\)[/tex])
The slope-intercept form of a line is:
[tex]\[
y = mx + b
\][/tex]
We need to solve for [tex]\(b\)[/tex]. Use one of the given points and the slope to solve for [tex]\(b\)[/tex]. Choose the point [tex]\((-5, -\frac{11}{5})\)[/tex]:
[tex]\[
y = \frac{2}{5}x + b
\][/tex]
Substitute [tex]\((-5, -\frac{11}{5})\)[/tex] into the equation:
[tex]\[
-\frac{11}{5} = \frac{2}{5}(-5) + b
\][/tex]
Simplify and solve for [tex]\(b\)[/tex]:
[tex]\[
-\frac{11}{5} = -2 + b
\][/tex]
[tex]\[
-\frac{11}{5} + 2 = b
\][/tex]
Convert 2 to a fraction with a denominator of 5:
[tex]\[
-\frac{11}{5} + \frac{10}{5} = b
\][/tex]
[tex]\[
-\frac{1}{5} = b
\][/tex]
This results in a y-intercept [tex]\(b = -\frac{1}{5}\)[/tex].
### Final Equation
Substitute the slope and y-intercept back into the slope-intercept form:
[tex]\[
y = \frac{2}{5}x - \frac{1}{5}
\][/tex]
Comparing with the given answer choices, the correct one is:
[tex]\[
y = \frac{2}{5}x - \frac{1}{5}
\][/tex]
Thus, the slope-intercept form of the equation for the line is:
[tex]\[
y = \frac{2}{5}x - \frac{1}{5}
\][/tex]
