To write the equation of the line in slope-intercept form, we begin with the given information:
- The slope [tex]\( m \)[/tex] is [tex]\(\frac{4}{5}\)[/tex].
- The point [tex]\((x_1, y_1)\)[/tex] through which the line passes is [tex]\((2, 7)\)[/tex].
The slope-intercept form of a line is given by:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
We also know the point-slope form of a line, which can be helpful for transitioning to the slope-intercept form:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting the given values, [tex]\((x_1, y_1) = (2, 7)\)[/tex] and [tex]\( m = \frac{4}{5}\)[/tex], we get:
[tex]\[ y - 7 = \frac{4}{5}(x - 2) \][/tex]
To convert this into the slope-intercept form, we need to solve for [tex]\( y \)[/tex]:
1. Expand the right-hand side:
[tex]\[ y - 7 = \frac{4}{5}x - \frac{4}{5} \cdot 2 \][/tex]
2. Simplify the expression:
[tex]\[ y - 7 = \frac{4}{5}x - \frac{8}{5} \][/tex]
3. Add [tex]\( 7 \)[/tex] to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{4}{5}x - \frac{8}{5} + 7 \][/tex]
To further simplify, convert 7 to a fraction with a denominator of 5:
[tex]\[ 7 = \frac{35}{5} \][/tex]
Thus, the equation becomes:
[tex]\[ y = \frac{4}{5}x - \frac{8}{5} + \frac{35}{5} \][/tex]
[tex]\[ y = \frac{4}{5}x + \frac{27}{5} \][/tex]
So, in slope-intercept form, the equation of the line is:
[tex]\[ y = 0.8x + 5.4 \][/tex]
Here, the slope [tex]\( m = 0.8 \)[/tex] and the y-intercept [tex]\( b = 5.4 \)[/tex]. Therefore, the final equation for the line is:
[tex]\[ y = 0.8x + 5.4 \][/tex]
