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What is the equation of a line that contains the point [tex]$(2,1)$[/tex] and is parallel to the line [tex]$y=3x-4$[/tex]?

A. [tex]y=3x-5[/tex]

B. [tex]y=3x-4[/tex]

C. [tex]y=-\frac{1}{3}x-4[/tex]

D. [tex]y=-\frac{1}{3}x-\frac{5}{3}[/tex]



Answer :

To determine the equation of a line that passes through the point [tex]\( (2,1) \)[/tex] and is parallel to the line [tex]\( y = 3x - 4 \)[/tex], let’s follow these steps:

1. Identify the slope of the given line:
The slope of the line [tex]\( y = 3x - 4 \)[/tex] is the coefficient of [tex]\( x \)[/tex] which is 3.

2. Use point-slope form of the line equation:
Since the new line must be parallel to [tex]\( y = 3x - 4 \)[/tex], it will have the same slope. The point-slope form of the equation of a line 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. Plugging in the given point [tex]\( (2,1) \)[/tex] and slope [tex]\( 3 \)[/tex]:
[tex]\[ y - 1 = 3(x - 2) \][/tex]

3. Simplify the equation:
Distribute the slope on the right side:
[tex]\[ y - 1 = 3x - 6 \][/tex]

Add 1 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 6 + 1 \][/tex]
[tex]\[ y = 3x - 5 \][/tex]

4. Identify the correct option:
The equation of the line that passes through [tex]\( (2, 1) \)[/tex] and is parallel to [tex]\( y = 3x - 4 \)[/tex] is [tex]\( y = 3x - 5 \)[/tex]. This corresponds to option A.

Thus, the answer is:
A. [tex]\( y = 3x - 5 \)[/tex]