To find the equation of a line that passes through a given point and has a given slope, we can use the point-slope form of the equation of a line. The point-slope form is written as:
[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 of the line.
Given:
- The point [tex]\((-1, 6)\)[/tex]
- The slope [tex]\(m = -3\)[/tex]
We substitute these values into the point-slope form equation:
[tex]\[ y - 6 = -3(x + 1) \][/tex]
Next, we simplify the equation to get it into slope-intercept form [tex]\(y = mx + b\)[/tex]:
1. Distribute the slope on the right-hand side:
[tex]\[ y - 6 = -3(x + 1) \][/tex]
[tex]\[ y - 6 = -3x - 3 \][/tex]
2. Add 6 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y = -3x - 3 + 6 \][/tex]
[tex]\[ y = -3x + 3 \][/tex]
So, the equation of the line in slope-intercept form is:
[tex]\[ y = -3x + 3 \][/tex]
From the provided options, the equation that represents the line is:
[tex]\[ y = -3x + 3 \][/tex]
