To find the equation of a line with a given slope and a point that it passes through, we can use the point-slope form of the line equation, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line.
Given:
- The slope [tex]\(m = \frac{1}{3}\)[/tex]
- The point is [tex]\((3, 8)\)[/tex]
Let's substitute these values into the point-slope form of the line equation.
1. Identify the values:
[tex]\[ m = \frac{1}{3} \][/tex]
[tex]\[ x_1 = 3 \][/tex]
[tex]\[ y_1 = 8 \][/tex]
2. Substitute into the point-slope form equation:
[tex]\[ y - 8 = \frac{1}{3}(x - 3) \][/tex]
3. Simplify the equation to slope-intercept form (y = mx + b):
First, distribute the slope [tex]\(\frac{1}{3}\)[/tex] across [tex]\((x - 3)\)[/tex]:
[tex]\[ y - 8 = \frac{1}{3}x - \frac{1}{3}(3) \][/tex]
[tex]\[ y - 8 = \frac{1}{3}x - 1 \][/tex]
4. Solve for [tex]\(y\)[/tex]:
Add 8 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y = \frac{1}{3}x - 1 + 8 \][/tex]
[tex]\[ y = \frac{1}{3}x + 7 \][/tex]
Therefore, the equation of the line is:
[tex]\[ y = \frac{1}{3}x + 7 \][/tex]
In summary, the equation of the line that has a slope of [tex]\(\frac{1}{3}\)[/tex] and passes through the point [tex]\((3, 8)\)[/tex] is:
[tex]\[ y = \frac{1}{3}x + 7 \][/tex]
