To find the equation of a line with a given slope and passing through a specific point, you can use the point-slope form of the equation of a line. The point-slope form is written as:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( m \)[/tex] is the slope of the line, and [tex]\( (x_1, y_1) \)[/tex] is a point on the line.
Given:
- Slope, [tex]\( m = 5 \)[/tex]
- Point, [tex]\( (x_1, y_1) = (4, 2) \)[/tex]
We plug these values into the point-slope form equation:
[tex]\[ y - 2 = 5(x - 4) \][/tex]
Now, we need to simplify this equation to get it into slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 2 = 5(x - 4) \][/tex]
[tex]\[ y - 2 = 5x - 20 \][/tex]
Next, we isolate [tex]\( y \)[/tex] by adding 2 to both sides of the equation:
[tex]\[ y = 5x - 20 + 2 \][/tex]
[tex]\[ y = 5x - 18 \][/tex]
In function notation, the equation of the line can be written as:
[tex]\[ f(x) = 5x - 18 \][/tex]
Therefore, the equation of the line is:
[tex]\[ f(x) = 5x - 18 \][/tex]
