Let's find the equation of the line that passes through the point [tex]\((2, -4)\)[/tex] and has a slope of [tex]\(-4\)[/tex].
To do this, we can use the point-slope form of a line, which 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. Given [tex]\((x_1, y_1) = (2, -4)\)[/tex] and [tex]\(m = -4\)[/tex], we can substitute these values into the point-slope form:
[tex]\[ y - (-4) = -4(x - 2) \][/tex]
Simplify the left side of the equation:
[tex]\[ y + 4 = -4(x - 2) \][/tex]
Next, we expand the right side:
[tex]\[ y + 4 = -4x + 8 \][/tex]
To get the equation into slope-intercept form [tex]\(y = mx + b\)[/tex], we need to isolate [tex]\(y\)[/tex]. Subtract 4 from both sides to achieve this:
[tex]\[ y = -4x + 8 - 4 \][/tex]
Simplify the expression on the right side:
[tex]\[ y = -4x + 4 \][/tex]
Therefore, the equation of the line that passes through the point [tex]\((2, -4)\)[/tex] and has a slope of [tex]\(-4\)[/tex] is:
[tex]\[ y = -4x + 4 \][/tex]
