To find the equation of the line with a given slope [tex]\( m = \frac{1}{5} \)[/tex] that passes through the point [tex]\((5, 4)\)[/tex], we can use the point-slope form of the equation of a line:
[tex]\[ y - y_1 = m(x - x1) \][/tex]
Substitute the given slope [tex]\( m = \frac{1}{5} \)[/tex] and the point [tex]\((x_1, y_1) = (5, 4)\)[/tex] into the point-slope form:
[tex]\[ y - 4 = \frac{1}{5}(x - 5) \][/tex]
Now, we can convert this equation into the slope-intercept form [tex]\( y = mx + b \)[/tex] where [tex]\( b \)[/tex] is the y-intercept. To do this, we should first distribute the slope [tex]\( \frac{1}{5} \)[/tex] on the right-hand side:
[tex]\[ y - 4 = \frac{1}{5}x - \frac{1}{5} \cdot 5 \][/tex]
This simplifies to:
[tex]\[ y - 4 = \frac{1}{5}x - 1 \][/tex]
Next, we isolate [tex]\( y \)[/tex] by adding 4 to both sides of the equation:
[tex]\[ y = \frac{1}{5}x - 1 + 4 \][/tex]
Simplify further to find the y-intercept:
[tex]\[ y = \frac{1}{5}x + 3 \][/tex]
Therefore, the equation of the line in slope-intercept form is:
[tex]\[ y = 0.2x + 3 \][/tex]
In this process, we've found that the slope is [tex]\( 0.2 \)[/tex] (which is equivalent to [tex]\( \frac{1}{5} \)[/tex]), and the y-intercept [tex]\( b \)[/tex] is [tex]\( 3.0 \)[/tex]. Thus, the complete equation of the line is:
[tex]\[ y = 0.2x + 3.0 \][/tex]
