Certainly! Let's derive the equation of the line that passes through the point [tex]\((-6, 8)\)[/tex] and has a slope of [tex]\(-\frac{5}{3}\)[/tex].
To find the equation of the line, we use 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]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope.
1. Substitute the given point [tex]\((-6, 8)\)[/tex] and the slope [tex]\(-\frac{5}{3}\)[/tex] into the point-slope form:
[tex]\[ y - 8 = -\frac{5}{3}(x + 6) \][/tex]
2. Distribute the slope [tex]\(-\frac{5}{3}\)[/tex] on the right-hand side:
[tex]\[ y - 8 = -\frac{5}{3}x - \frac{5}{3} \cdot 6 \][/tex]
3. Calculate [tex]\(-\frac{5}{3} \cdot 6\)[/tex]:
[tex]\[ -\frac{5}{3} \cdot 6 = -10 \][/tex]
So the equation becomes:
[tex]\[ y - 8 = -\frac{5}{3}x - 10 \][/tex]
4. To convert this into the slope-intercept form [tex]\( y = mx + b \)[/tex], solve for [tex]\( y \)[/tex] by adding 8 to both sides:
[tex]\[ y = -\frac{5}{3}x - 10 + 8 \][/tex]
5. Combine like terms on the right-hand side:
[tex]\[ y = -\frac{5}{3}x - 2 \][/tex]
So, the equation of the line in slope-intercept form is:
[tex]\[ y = -\frac{5}{3}x - 2 \][/tex]
Therefore, the equation of the line that passes through the point [tex]\((-6, 8)\)[/tex] and has a slope of [tex]\(-\frac{5}{3}\)[/tex] is:
[tex]\[ y = -\frac{5}{3}x - 2 \][/tex]
