To find the equation of the line passing through the point [tex]\((2, 3)\)[/tex] with a slope of [tex]\(-6\)[/tex], we use the slope-intercept form of a line, which is given by:
[tex]\[ y = mx + b \][/tex]
Here, [tex]\(m\)[/tex] represents the slope and [tex]\(b\)[/tex] is the y-intercept.
Steps to find the equation:
1. Identify the slope ([tex]\(m\)[/tex]): The slope of the line is given as [tex]\(-6\)[/tex].
2. Identify a point on the line [tex]\((x_1, y_1)\)[/tex]: The point given is [tex]\((2, 3)\)[/tex].
3. Substitute the point and slope into the slope-intercept equation: We know the point [tex]\((x_1, y_1)\)[/tex] satisfies the equation of the line. Therefore, we can substitute [tex]\(x_1 = 2\)[/tex], [tex]\(y_1 = 3\)[/tex], and [tex]\(m = -6\)[/tex] into the equation to find [tex]\(b\)[/tex], the y-intercept.
[tex]\[ y_1 = mx_1 + b \][/tex]
[tex]\[ 3 = -6(2) + b \][/tex]
[tex]\[ 3 = -12 + b \][/tex]
4. Solve for [tex]\(b\)[/tex] (the y-intercept):
[tex]\[ 3 = -12 + b \][/tex]
[tex]\[ b = 3 + 12 \][/tex]
[tex]\[ b = 15 \][/tex]
5. Write the equation: Now that we have the slope ([tex]\(m\)[/tex]) and the y-intercept ([tex]\(b\)[/tex]), we can write the equation of the line:
[tex]\[ y = mx + b \][/tex]
[tex]\[ y = -6x + 15 \][/tex]
Thus, the equation in slope-intercept form for the line that passes through the point [tex]\((2, 3)\)[/tex] and has a slope of [tex]\(-6\)[/tex] is:
[tex]\[ y = -6x + 15 \][/tex]
