Sure, let's solve each part step-by-step:
### Part (a)
We need to find the equation of a line with a slope of [tex]\(-\frac{1}{2}\)[/tex] that passes through the point [tex]\((5, -2)\)[/tex]. We will use the point-slope form of the equation of a line:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line. Here, [tex]\(m = -\frac{1}{2}\)[/tex], [tex]\(x_1 = 5\)[/tex], and [tex]\(y_1 = -2\)[/tex].
Substituting these values into the formula, we get:
[tex]\[ y - (-2) = -\frac{1}{2}(x - 5) \][/tex]
[tex]\[ y + 2 = -\frac{1}{2}x + \frac{5}{2} \][/tex]
Now, solve for [tex]\(y\)[/tex] to get it into slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y = -\frac{1}{2}x + \frac{5}{2} - 2 \][/tex]
[tex]\[ y = -\frac{1}{2}x + \frac{5}{2} - \frac{4}{2} \][/tex]
[tex]\[ y = -\frac{1}{2}x + \frac{1}{2} \][/tex]
So, the equation of the line is:
[tex]\[ y = -\frac{1}{2}x + \frac{1}{2} \][/tex]
### Part (b)
We need to find the equation of a line that passes through the points [tex]\((1, 4)\)[/tex] and [tex]\((3, 0)\)[/tex]. First, calculate the slope [tex]\(m\)[/tex] using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Taking [tex]\((x_1, y_1) = (1, 4)\)[/tex] and [tex]\((x_2, y_2) = (3, 0)\)[/tex]:
[tex]\[ m = \frac{0 - 4}{3 - 1} \][/tex]
[tex]\[ m = \frac{-4}{2} \][/tex]
[tex]\[ m = -2 \][/tex]
Next, use the point-slope form of the equation of a line with the slope [tex]\(m = -2\)[/tex] and one of the given points, say [tex]\((1, 4)\)[/tex]:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
[tex]\[ y - 4 = -2(x - 1) \][/tex]
Solve for [tex]\(y\)[/tex] to get it into slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y - 4 = -2x + 2 \][/tex]
[tex]\[ y = -2x + 2 + 4 \][/tex]
[tex]\[ y = -2x + 6 \][/tex]
So, the equation of the line is:
[tex]\[ y = -2x + 6 \][/tex]
### Summary
The equations of the lines are:
a. [tex]\( y = -\frac{1}{2}x + \frac{1}{2} \)[/tex]
b. [tex]\( y = -2x + 6 \)[/tex]
