To find the equation of a line with a given gradient and a point it passes through, we can use the point-slope form of the line equation. The point-slope form is given as:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where:
- \( (x_1, y_1) \) is a point on the line.
- \( m \) is the gradient (slope) of the line.
Given:
- The gradient \( m = \frac{1}{2} \)
- The point \( (4, -2) \)
We can substitute these values into the point-slope form equation:
[tex]\[ y - (-2) = \frac{1}{2}(x - 4) \][/tex]
This simplifies to:
[tex]\[ y + 2 = \frac{1}{2}(x - 4) \][/tex]
Next, we need to simplify this equation further to get it into the slope-intercept form \( y = mx + c \).
First, distribute the gradient \( \frac{1}{2} \) on the right side:
[tex]\[ y + 2 = \frac{1}{2}x - \frac{1}{2} \times 4 \][/tex]
[tex]\[ y + 2 = \frac{1}{2}x - 2 \][/tex]
Then, isolate \( y \) by subtracting 2 from both sides of the equation:
[tex]\[ y = \frac{1}{2}x - 2 - 2 \][/tex]
[tex]\[ y = \frac{1}{2}x - 4 \][/tex]
Therefore, the equation of the line with a gradient of \( \frac{1}{2} \) that passes through the point \( (4, -2) \) is:
[tex]\[ y = \frac{1}{2}x - 4 \][/tex]
