To determine the equation of the line that has a slope of [tex]\(\frac{1}{3}\)[/tex] and passes through the point [tex]\((-2, 1)\)[/tex], you can use the slope-intercept form of a line, which is [tex]\(y = mx + b\)[/tex]. Here, [tex]\(m\)[/tex] represents the slope, and [tex]\(b\)[/tex] represents the y-intercept.
Given:
- Slope ([tex]\(m\)[/tex]) = [tex]\(\frac{1}{3}\)[/tex]
- Point [tex]\((-2, 1)\)[/tex] which lies on the line
Here's the step-by-step process:
1. Substitute the given slope and the coordinates of the point into the equation [tex]\(y = mx + b\)[/tex] to solve for [tex]\(b\)[/tex].
[tex]\[
y = mx + b
\][/tex]
2. Plug in the values [tex]\(x = -2\)[/tex] and [tex]\(y = 1\)[/tex] along with the slope [tex]\(m = \frac{1}{3}\)[/tex].
[tex]\[
1 = \left(\frac{1}{3}\right)(-2) + b
\][/tex]
3. Simplify the equation to find [tex]\(b\)[/tex].
[tex]\[
1 = -\frac{2}{3} + b
\][/tex]
4. To isolate [tex]\(b\)[/tex], add [tex]\(\frac{2}{3}\)[/tex] to both sides of the equation.
[tex]\[
1 + \frac{2}{3} = b
\][/tex]
5. Change 1 to a fraction with a common denominator of 3.
[tex]\[
\frac{3}{3} + \frac{2}{3} = b
\][/tex]
6. Add the fractions on the left-hand side.
[tex]\[
\frac{3}{3} + \frac{2}{3} = \frac{5}{3}
\][/tex]
Therefore:
[tex]\[
b = \frac{5}{3}
\][/tex]
So, the y-intercept [tex]\(b\)[/tex] is [tex]\(\frac{5}{3}\)[/tex].
7. Substitute the values of [tex]\(m\)[/tex] and [tex]\(b\)[/tex] back into the slope-intercept equation [tex]\(y = mx + b\)[/tex].
[tex]\[
y = \left(\frac{1}{3}\right)x + \frac{5}{3}
\][/tex]
Hence, the equation of the line in slope-intercept form is:
[tex]\[
y = \frac{1}{3}x + \frac{5}{3}
\][/tex]
