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What is the equation of the line that is parallel to the given line and passes through the point [tex]\((-2, 2)\)[/tex]?

A. [tex]\(y = \frac{1}{5} x + 4\)[/tex]
B. [tex]\(y = \frac{1}{5} x + \frac{12}{5}\)[/tex]
C. [tex]\(y = -5 x + 4\)[/tex]
D. [tex]\(y = -5 x + \frac{12}{5}\)[/tex]


Sagot :

To determine the equation of a line that is parallel to the given line and passes through the specified point, we follow these steps:

1. Identify the slope of the given line: The given line is [tex]\( y = \frac{1}{5} x + 4 \)[/tex]. The slope [tex]\( m \)[/tex] of this line is [tex]\( \frac{1}{5} \)[/tex].

2. Remember the property of parallel lines: If two lines are parallel, their slopes are equal. Hence, the slope of the line we are looking for is also [tex]\( \frac{1}{5} \)[/tex].

3. Use the point-slope form of the equation of a line: The point-slope form is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope. Here, the point is [tex]\( (-2, 2) \)[/tex] and the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{5} \)[/tex].

4. Substitute the values into the point-slope form:
[tex]\[ y - 2 = \frac{1}{5}(x - (-2)) \][/tex]

5. Simplify the equation:
[tex]\[ y - 2 = \frac{1}{5}(x + 2) \][/tex]
[tex]\[ y - 2 = \frac{1}{5}x + \frac{1}{5} \cdot 2 \][/tex]
[tex]\[ y - 2 = \frac{1}{5}x + \frac{2}{5} \][/tex]

6. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{5}x + \frac{2}{5} + 2 \][/tex]
Convert the constant term to a common denominator:
[tex]\[ y = \frac{1}{5}x + \frac{2}{5} + \frac{10}{5} \][/tex]
[tex]\[ y = \frac{1}{5}x + \frac{12}{5} \][/tex]

Therefore, the equation of the line that is parallel to the given line and passes through the point [tex]\((-2, 2)\)[/tex] is [tex]\( y = \frac{1}{5}x + \frac{12}{5} \)[/tex].

Among the provided options, the correct one is:
[tex]\[ y = \frac{1}{5} x + \frac{12}{5} \][/tex]