To determine the equation of a line that has a gradient (slope) of -3 and passes through the point [tex]\((-1, 4)\)[/tex], we can use the point-slope form of the linear equation. The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( m \)[/tex] is the slope, and [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes.
Here, we are given:
- The slope [tex]\( m = -3 \)[/tex]
- The point [tex]\((x_1, y_1) = (-1, 4)\)[/tex]
Let's substitute these values into the point-slope form equation:
[tex]\[ y - 4 = -3(x - (-1)) \][/tex]
Simplify the equation inside the parentheses:
[tex]\[ y - 4 = -3(x + 1) \][/tex]
Next, we'll distribute the slope [tex]\(-3\)[/tex] on the right side:
[tex]\[ y - 4 = -3x - 3 \][/tex]
To get the equation into the slope-intercept form [tex]\( y = mx + c \)[/tex], we'll solve for [tex]\( y \)[/tex]. To do this, we'll add 4 to both sides of the equation:
[tex]\[ y = -3x - 3 + 4 \][/tex]
Combine the constant terms on the right side:
[tex]\[ y = -3x + 1 \][/tex]
Thus, the equation of the line with a gradient of -3 passing through the point [tex]\((-1, 4)\)[/tex] is:
[tex]\[ y = -3x + 1 \][/tex]
