To find the equation of a line with a given slope and a point through which it passes, we can use the point-slope form of a linear equation. The point-slope form is given by:
[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:
- Slope ([tex]\( m \)[/tex]) = 4
- Point ([tex]\( x_1, y_1 \)[/tex]) = (1, 6)
Substitute the given slope and point into the point-slope form equation:
[tex]\[ y - 6 = 4(x - 1) \][/tex]
Now, we'll solve for [tex]\( y \)[/tex]:
1. Distribute the slope [tex]\( 4 \)[/tex] on the right-hand side:
[tex]\[ y - 6 = 4x - 4 \][/tex]
2. Add 6 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 4x - 4 + 6 \][/tex]
[tex]\[ y = 4x + 2 \][/tex]
So, the equation of the line is [tex]\( y = 4x + 2 \)[/tex].
Now, let's examine the given options to determine which one matches this equation:
1. [tex]\( y = 4x - 2 \)[/tex] (y-intercept = -2)
2. [tex]\( y = 4x + 6 \)[/tex] (y-intercept = 6)
3. [tex]\( y = 4x + 2 \)[/tex] (y-intercept = 2)
4. [tex]\( y = 4x - 3 \)[/tex] (y-intercept = -3)
The equation [tex]\( y = 4x + 2 \)[/tex] corresponds to option 3.
Therefore, the correct equation of the line is:
[tex]\[ \boxed{y = 4x + 2} \][/tex]
