To find the equation of a line that passes through the point [tex]\((11, 11)\)[/tex] with a slope of [tex]\(-\frac{1}{2}\)[/tex], we can start by using the point-slope form of the equation of a line. The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\(m\)[/tex] is the slope of the line. Given the point [tex]\((11, 11)\)[/tex] and the slope [tex]\(m = -\frac{1}{2}\)[/tex], we can substitute these values into the point-slope form:
[tex]\[ y - 11 = -\frac{1}{2} (x - 11) \][/tex]
Next, we need to convert this equation to the slope-intercept form, which is [tex]\(y = mx + b\)[/tex]. To do this, we will first distribute the slope [tex]\(-\frac{1}{2}\)[/tex] on the right-hand side of the equation:
[tex]\[ y - 11 = -\frac{1}{2} \cdot x + (-\frac{1}{2} \cdot 11) \][/tex]
[tex]\[ y - 11 = -\frac{1}{2}x - \frac{11}{2} \][/tex]
Now we add 11 to both sides of the equation to isolate [tex]\(y\)[/tex] on the left-hand side:
[tex]\[ y = -\frac{1}{2}x - \frac{11}{2} + 11 \][/tex]
To combine the constants on the right-hand side, note that 11 is equivalent to [tex]\(\frac{22}{2}\)[/tex]:
[tex]\[ y = -\frac{1}{2} x - \frac{11}{2} + \frac{22}{2} \][/tex]
[tex]\[ y = -\frac{1}{2}x + \frac{11}{2} \][/tex]
As a result, the equation of the line in slope-intercept form is:
[tex]\[ y = -\frac{1}{2}x + \frac{33}{2} \][/tex]
Therefore, the equation of the line passing through the point [tex]\((11, 11)\)[/tex] with a slope of [tex]\(-\frac{1}{2}\)[/tex] is:
[tex]\[ y = -\frac{1}{2}x + \frac{33}{2} \][/tex]
