Let's analyze each of the given equations to determine if they could represent the line [tex]\( f(x) = 4x + 3 \)[/tex] that passes through the point [tex]\((1,7)\)[/tex].
### 1. Equation: [tex]\( y - 7 = 3(x - 1) \)[/tex]
First, we need to rewrite this equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]):
1. Distribute the 3 on the right-hand side:
[tex]\[
y - 7 = 3(x - 1) \\
y - 7 = 3x - 3
\][/tex]
2. Add 7 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 3x - 3 + 7 \\
y = 3x + 4
\][/tex]
The slope ([tex]\( m \)[/tex]) is 3 and the y-intercept is 4. This does not match our original line equation [tex]\( f(x) = 4x + 3 \)[/tex]. So, the first equation does not represent the same line.
### 2. Equation: [tex]\( y - 1 = 3(x - 7) \)[/tex]
Next, we transform this equation to slope-intercept form:
1. Distribute the 3 on the right-hand side:
[tex]\[
y - 1 = 3(x - 7) \\
y - 1 = 3x - 21
\][/tex]
2. Add 1 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 3x - 21 + 1 \\
y = 3x - 20
\][/tex]
The slope ([tex]\( m \)[/tex]) here is 3 and the y-intercept is -20. This also does not match our original line equation. Thus, the second equation does not represent the same line.
### 3. Equation: [tex]\( y - 7 = 4(x - 1) \)[/tex]
Let’s rewrite this equation in slope-intercept form:
1. Distribute the 4 on the right-hand side:
[tex]\[
y - 7 = 4(x - 1) \\
y - 7 = 4x - 4
\][/tex]
2. Add 7 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 4x - 4 + 7 \\
y = 4x + 3
\][/tex]
The slope ([tex]\( m \)[/tex]) here is 4 and the y-intercept is 3. This matches our original line equation [tex]\( f(x) = 4x + 3 \)[/tex]. Therefore, the third equation represents the same line.
### 4. Equation: [tex]\( y - 1 = 4(x - 7) \)[/tex]
Finally, transform this to slope-intercept form:
1. Distribute the 4 on the right-hand side:
[tex]\[
y - 1 = 4(x - 7) \\
y - 1 = 4x - 28
\][/tex]
2. Add 1 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 4x - 28 + 1 \\
y = 4x - 27
\][/tex]
The slope ([tex]\( m \)[/tex]) is 4 but the y-intercept is -27. This does not match our original line's intercept. Thus, the fourth equation does not represent the same line.
### Conclusion
Out of the four given equations, only [tex]\( y - 7 = 4(x - 1) \)[/tex] represents the same line as [tex]\( f(x) = 4x + 3 \)[/tex].
