To determine which equation could represent the same line as [tex]\( f(x) = 4x + 3 \)[/tex] and contains the point [tex]\((1, 7)\)[/tex], we need to use the point-slope form of a linear equation, which is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope of the line.
1. Identify the given point and slope:
- Given point: [tex]\( (1, 7) \)[/tex]
- Slope [tex]\( m \)[/tex] derived from the line [tex]\( f(x) = 4x + 3 \)[/tex] is [tex]\( 4 \)[/tex]
2. Substitute these values into the point-slope form:
- Plugging in [tex]\( (x_1, y_1) = (1, 7) \)[/tex] and [tex]\( m = 4 \)[/tex]:
[tex]\[ y - 7 = 4(x - 1) \][/tex]
So, the correct equation in point-slope form for the line that Chin wrote is:
[tex]\[ y - 7 = 4(x - 1) \][/tex]
3. Compare this derived equation with the given options:
- [tex]\( y - 7 = 3(x - 1) \)[/tex]:
- This equation has the wrong slope (3 instead of 4). It does not represent the same line.
- [tex]\( y - 1 = 3(x - 7) \)[/tex]:
- This equation also has the wrong slope (3 instead of 4) and uses the wrong point (making it even further off). It does not represent the same line.
- [tex]\( y - 7 = 4(x - 1) \)[/tex]:
- This equation is the same as the one we derived. It has the correct slope and correctly uses the given point.
- [tex]\( y - 1 = 4(x - 7) \)[/tex]:
- This equation has the correct slope (4) but uses the wrong point. It does not represent the same line.
From these comparisons, we find that the correct equation which represents the same line as [tex]\( f(x) = 4x + 3 \)[/tex] and contains the point [tex]\( (1, 7) \)[/tex] is:
[tex]\[ y - 7 = 4(x - 1) \][/tex]
So the correct option is the third one.
Therefore, the correct option is:
[tex]\[ \boxed{3} \][/tex]
