To find the equation of a line that passes through a given point [tex]\((x_1, y_1)\)[/tex] and has a specific slope [tex]\(m\)[/tex], we can 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]
Given the point [tex]\((-5, 1)\)[/tex] and the slope [tex]\(m = -3\)[/tex], we can substitute these values into the point-slope form equation:
[tex]\[ y - 1 = -3(x - (-5)) \][/tex]
Simplify the equation by replacing the double negative:
[tex]\[ y - 1 = -3(x + 5) \][/tex]
Now, distribute the slope [tex]\(-3\)[/tex] through the parentheses:
[tex]\[ y - 1 = -3x - 15 \][/tex]
Next, to get the equation into the slope-intercept form [tex]\(y = mx + b\)[/tex], solve for [tex]\(y\)[/tex] by adding 1 to both sides:
[tex]\[ y = -3x - 15 + 1 \][/tex]
Combine the constant terms on the right-hand side:
[tex]\[ y = -3x - 14 \][/tex]
So, the equation of the line is:
[tex]\[ y = -3x - 14 \][/tex]
