To find the equation of the line [tex]\(PQ\)[/tex] given the point [tex]\(P(3, 1)\)[/tex] and the gradient [tex]\(-\frac{1}{2}\)[/tex], we will use the point-slope form of the equation of a line.
The point-slope form of a line is given by:
[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 gradient (slope).
Here, we have:
- [tex]\( (x_1, y_1) = (3, 1) \)[/tex]
- [tex]\( m = -\frac{1}{2} \)[/tex]
Substituting these values into the point-slope form equation, we get:
[tex]\[ y - 1 = -\frac{1}{2}(x - 3) \][/tex]
Next, we simplify this equation to get it into the slope-intercept form [tex]\( y = mx + c \)[/tex], where [tex]\(m\)[/tex] is the gradient and [tex]\(c\)[/tex] is the y-intercept.
Start by distributing the gradient on the right-hand side:
[tex]\[ y - 1 = -\frac{1}{2}x + \frac{3}{2} \][/tex]
Next, isolate [tex]\(y\)[/tex] by adding 1 to both sides of the equation:
[tex]\[ y = -\frac{1}{2}x + \frac{3}{2} + 1 \][/tex]
Convert 1 to a fraction with a common denominator for easier addition:
[tex]\[ 1 = \frac{2}{2} \][/tex]
So now we have:
[tex]\[ y = -\frac{1}{2}x + \frac{3}{2} + \frac{2}{2} \][/tex]
Combine the fractions on the right-hand side:
[tex]\[ y = -\frac{1}{2}x + \frac{5}{2} \][/tex]
Thus, the equation of the line [tex]\(PQ\)[/tex] is:
[tex]\[ y = -\frac{1}{2}x + \frac{5}{2} \][/tex]
