Which equation represents a line that passes through \((-2, 4)\) and has a slope of \(\frac{2}{5}\)?

A. \(y - 4 = \frac{2}{5}(x + 2)\)

B. \(y + 4 = \frac{2}{5}(x - 2)\)

C. \(y + 2 = \frac{2}{5}(x - 4)\)

D. [tex]\(y - 2 = \frac{2}{5}(x + 4)\)[/tex]



Answer :

To determine which equation represents a line that passes through the point \((-2,4)\) and has a slope of \(\frac{2}{5}\), we can use the point-slope form of the equation of a line. The point-slope form is given by the equation:

[tex]\[ y - y_1 = m(x - x_1) \][/tex]

where \((x_1, y_1)\) is a point on the line and \(m\) is the slope of the line.

Given:
- The point \((-2, 4)\): \(x_1 = -2\), \(y_1 = 4\)
- The slope \(m = \frac{2}{5}\)

Substitute these values into the point-slope form equation:

[tex]\[ y - 4 = \frac{2}{5}(x - (-2)) \][/tex]

Simplify the equation:

[tex]\[ y - 4 = \frac{2}{5}(x + 2) \][/tex]

Now, let's compare this with the given options:

1. \(y - 4 = \frac{2}{5}(x + 2)\): This matches our derived equation exactly.

2. \(y + 4 = \frac{2}{5}(x - 2)\): This does not match our derived equation.

3. \(y + 2 = \frac{2}{5}(x - 4)\): This does not match our derived equation.

4. \(y - 2 = \frac{2}{5}(x + 4)\): This does not match our derived equation.

Thus, the correct equation that represents a line passing through the point \((-2, 4)\) with a slope of \(\frac{2}{5}\) is:

[tex]\[ y - 4 = \frac{2}{5}(x + 2) \][/tex]

So, the correct option is:

[tex]\[ \boxed{1} \][/tex]