Find an equation of the line that satisfies the given conditions:
through (2, 4), perpendicular to
x − 4y + 7 = 0.
To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line first. The given line is in the standard form: x − 4y + 7 = 0. To find the slope, we need to rewrite the equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
To convert the equation x − 4y + 7 = 0 into slope-intercept form, we isolate the y variable:
-4y = -x - 7
y = (1/4)x + (7/4)
Now we can determine the slope of the given line. In slope-intercept form, the coefficient of x (1/4) is the slope, so the slope of the given line is 1/4.
To find the slope of the line perpendicular to the given line, we use the fact that perpendicular lines have slopes that are negative reciprocals. Thus, the slope of the perpendicular line is -4/1, which simplifies to -4.
Now we have the slope (-4) and a point (2, 4) that the line passes through. We can use the point-slope form of a linear equation to find the equation of the line:
y - y1 = m(x - x1)
Substituting the values:
y - 4 = -4(x - 2)
Next, distribute -4 on the right side:
y - 4 = -4x + 8
Further rearranging the equation by isolating y:
y = -4x + 8 + 4
y = -4x + 12
Therefore, the equation of the line that passes through (2, 4) and is perpendicular to the line x − 4y + 7 = 0 is y = -4x + 12.