Find an equation of the line that satisfies the given conditions.
through (2, 4) perpendicular to
x − 4y + 7 = 0
I know that you would set it up as
-4y = -x - 7, but would you divide the 4 to get slope = (1/4)x, which would make the perpendicular slope 4?
Slopes of perpendicular lines are negative reciprocals of each other. One is + , the other is - .
Here is a little trick for perpendicular lines
If we have some line
Ax + By + C = 0
then the line
Bx -Ay + k = 0 is perpendicular, that is just switch the coefficients and make the sign opposite. The constant will probably be different.
So for yours, the new equation must be
4x + y + k = 0
plug in the given point
4(2) + 4 + k = 0
k = -12
4x + y - 12 = 0
now wasn't that easy ?
Took about 3 lines.
To find an equation of a line that is perpendicular to another line, you need to find the negative reciprocal of the slope of the given line.
The given line, x − 4y + 7 = 0, can also be written as -4y = -x - 7. Dividing both sides of the equation by -4 gives you y = (1/4)x + 7/4.
So, the slope of the given line is (1/4).
To find the slope of the line perpendicular to it, you need to take the negative reciprocal of (1/4). The negative reciprocal of (1/4) is -4.
Therefore, the perpendicular slope is -4.
Now, you can use the point-slope form of a linear equation, y - y₁ = m(x - x₁), where (x₁, y₁) represents the point (2, 4) and m represents the perpendicular slope (-4).
Plugging in the values, the equation becomes: y - 4 = -4(x - 2).
Simplifying further, you get: y - 4 = -4x + 8.
To obtain the standard form of the equation, move all terms to the left side of the equation: 4x + y = 12.
Therefore, the equation of the line that is perpendicular to x − 4y + 7 = 0 and passes through the point (2, 4) is 4x + y = 12.
To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
Given the equation x - 4y + 7 = 0, we can rearrange it to the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. Let's solve for y:
-4y = -x - 7
Divide both sides by -4 to isolate y:
y = (1/4)x + 7/4
The slope of the given line is 1/4. To find the slope of the line that is perpendicular to it, we take the negative reciprocal of 1/4.
The negative reciprocal of 1/4 is -4. Therefore, the slope of the line that is perpendicular to x - 4y + 7 = 0 is -4.
Now that we know the slope (-4) and a point (2, 4) on the line, we can use the point-slope form of the equation of a line to find the equation:
y - y1 = m(x - x1)
Where (x1, y1) is the given point and m is the slope.
Plugging in the values:
y - 4 = -4(x - 2)
Expanding and simplifying:
y - 4 = -4x + 8
y = -4x + 12
Therefore, the equation of the line that passes through (2, 4) and is perpendicular to x - 4y + 7 = 0 is y = -4x + 12.