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Sagot :
To determine which of the provided equations represents the same line as [tex]\( f(x) = 4x + 3 \)[/tex], we need to analyze the given equations for consistency with the slope and a point on the line.
Chin's equation is:
[tex]\[ f(x) = 4x + 3 \][/tex]
From this equation, we can see that:
- The slope ([tex]\(m\)[/tex]) is [tex]\(4\)[/tex].
- The line passes through the point [tex]\((1, 7)\)[/tex] because substituting [tex]\(x = 1\)[/tex] into the equation yields:
[tex]\[ f(1) = 4(1) + 3 = 7 \][/tex]
Thus, the point [tex]\((1, 7)\)[/tex] lies on the line.
Now let's analyze each of the given equations:
1. [tex]\( y - 7 = 3(x - 1) \)[/tex]
- This is in point-slope form, [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, the slope [tex]\(m\)[/tex] is [tex]\(3\)[/tex] and the line goes through the point [tex]\((1, 7)\)[/tex].
- Since the slope [tex]\(3\)[/tex] does not match the slope [tex]\(4\)[/tex] of Chin's equation, this equation does not represent the same line.
2. [tex]\( y - 1 = 3(x - 7) \)[/tex]
- This is in point-slope form, [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, the slope [tex]\(m\)[/tex] is [tex]\(3\)[/tex] and the line goes through the point [tex]\((7, 1)\)[/tex].
- Neither the slope [tex]\(3\)[/tex] matches [tex]\(4\)[/tex] nor the point [tex]\((7, 1)\)[/tex] lies on the line [tex]\(f(x) = 4x + 3\)[/tex], so this equation does not represent the same line.
3. [tex]\( y - 7 = 4(x - 1) \)[/tex]
- This is in point-slope form, [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, the slope [tex]\(m\)[/tex] is [tex]\(4\)[/tex] and the line goes through the point [tex]\((1, 7)\)[/tex].
- Both the slope [tex]\(4\)[/tex] and the point [tex]\((1, 7)\)[/tex] match Chin’s line, so this equation correctly represents the same line.
4. [tex]\( y - 1 = 4(x - 7) \)[/tex]
- This is in point-slope form, [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, the slope [tex]\(m\)[/tex] is [tex]\(4\)[/tex] and the line goes through the point [tex]\((7, 1)\)[/tex].
- While the slope [tex]\(4\)[/tex] matches, the point [tex]\((7, 1)\)[/tex] is not on the line described by [tex]\(f(x) = 4x + 3\)[/tex], so this equation does not represent the same line.
Thus, the equation that could represent the same line is:
[tex]\[ y - 7 = 4(x - 1) \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{3} \][/tex]
Chin's equation is:
[tex]\[ f(x) = 4x + 3 \][/tex]
From this equation, we can see that:
- The slope ([tex]\(m\)[/tex]) is [tex]\(4\)[/tex].
- The line passes through the point [tex]\((1, 7)\)[/tex] because substituting [tex]\(x = 1\)[/tex] into the equation yields:
[tex]\[ f(1) = 4(1) + 3 = 7 \][/tex]
Thus, the point [tex]\((1, 7)\)[/tex] lies on the line.
Now let's analyze each of the given equations:
1. [tex]\( y - 7 = 3(x - 1) \)[/tex]
- This is in point-slope form, [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, the slope [tex]\(m\)[/tex] is [tex]\(3\)[/tex] and the line goes through the point [tex]\((1, 7)\)[/tex].
- Since the slope [tex]\(3\)[/tex] does not match the slope [tex]\(4\)[/tex] of Chin's equation, this equation does not represent the same line.
2. [tex]\( y - 1 = 3(x - 7) \)[/tex]
- This is in point-slope form, [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, the slope [tex]\(m\)[/tex] is [tex]\(3\)[/tex] and the line goes through the point [tex]\((7, 1)\)[/tex].
- Neither the slope [tex]\(3\)[/tex] matches [tex]\(4\)[/tex] nor the point [tex]\((7, 1)\)[/tex] lies on the line [tex]\(f(x) = 4x + 3\)[/tex], so this equation does not represent the same line.
3. [tex]\( y - 7 = 4(x - 1) \)[/tex]
- This is in point-slope form, [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, the slope [tex]\(m\)[/tex] is [tex]\(4\)[/tex] and the line goes through the point [tex]\((1, 7)\)[/tex].
- Both the slope [tex]\(4\)[/tex] and the point [tex]\((1, 7)\)[/tex] match Chin’s line, so this equation correctly represents the same line.
4. [tex]\( y - 1 = 4(x - 7) \)[/tex]
- This is in point-slope form, [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, the slope [tex]\(m\)[/tex] is [tex]\(4\)[/tex] and the line goes through the point [tex]\((7, 1)\)[/tex].
- While the slope [tex]\(4\)[/tex] matches, the point [tex]\((7, 1)\)[/tex] is not on the line described by [tex]\(f(x) = 4x + 3\)[/tex], so this equation does not represent the same line.
Thus, the equation that could represent the same line is:
[tex]\[ y - 7 = 4(x - 1) \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{3} \][/tex]
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