To determine the equation of a line that passes through the point [tex]\((2, 1)\)[/tex] and is parallel to the line given by the equation [tex]\(y = 3x - 4\)[/tex], follow these steps:
1. Identify the slope of the given line:
The equation provided is [tex]\(y = 3x - 4\)[/tex]. In the slope-intercept form of a linear equation, which is [tex]\(y = mx + b\)[/tex], [tex]\(m\)[/tex] represents the slope. Therefore, the slope [tex]\(m\)[/tex] of the given line is 3.
2. Understanding parallel lines:
Lines that are parallel to each other have identical slopes. Hence, any line that is parallel to the given line will also have a slope of 3.
3. Point-slope form:
To find the equation of the line that passes through a specified point with a given slope, use the point-slope form of a linear equation, which is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes and [tex]\(m\)[/tex] is the slope. Here, the point is [tex]\((2, 1)\)[/tex], so [tex]\(x_1 = 2\)[/tex] and [tex]\(y_1 = 1\)[/tex], and the slope [tex]\(m = 3\)[/tex].
4. Substitute the values into the point-slope form:
[tex]\[
y - 1 = 3(x - 2)
\][/tex]
5. Simplify the equation:
First, distribute the slope on the right side:
[tex]\[
y - 1 = 3x - 6
\][/tex]
To convert this to slope-intercept form, add 1 to both sides:
[tex]\[
y = 3x - 6 + 1
\][/tex]
Combine the constants:
[tex]\[
y = 3x - 5
\][/tex]
6. Identify the correct option:
The equation of the line that passes through the point [tex]\((2, 1)\)[/tex] and is parallel to the given line is [tex]\(y = 3x - 5\)[/tex]. This corresponds to option A.
Therefore, the correct answer is:
A. [tex]\(y = 3x - 5\)[/tex]
