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]
[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]
