Find an equation of the line L that passes through the point
(−2, 6)
and satisfies the given condition. (Let x be the independent variable and y be the dependent variable.)
L is perpendicular to the line
4x + 5y = 10
given line: 4x + 5y = 10 ----> slope = -4/5
so the slope of your new line is 5/4
and its equation must be
5x - 4y = C
plug in (−2, 6), find C
and you are done
perpendicular to 4x + 5 y = 10
well what slope
5 y = -4 x + 10
m = -4/5
so we want slope perpendicular to that
m' = -1/m = 5/4
so our line is
y = (5/4) x + b
so what is b ?
put the point in
6 = (5/4)(-2) + b
6 * 4 = -10 + 4 b
4 b = 24+ 10 = 34
b = 34/2
so our linne is
y = (5/4) x + 34/2
or
4 y = 5 x + 68
4 y - 5 x = 68
check my arithmetic, going fast
is the answer 5x-4y=-34 ?
yes, I have an arithmetic error
4 b = 24+ 10 = 34
b = 34/4
so our linne is
y = (5/4) x + 34/4
or
4 y = 5 x + 34
4 y - 5 x = 34
or
5 x - 4 y = -34
To find the equation of a line that is perpendicular to a given line, we need to first determine the slope of the given line.
The given line is 4x + 5y = 10. Let's rearrange this equation to the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
Starting with the given equation:
4x + 5y = 10
Subtract 4x from both sides:
5y = -4x + 10
Divide both sides by 5:
y = (-4/5)x + 2
From this equation, we can see that the slope of the given line is -4/5.
To find the slope of the line perpendicular to this one, we need to take the negative reciprocal of -4/5.
The negative reciprocal of -4/5 is 5/4.
Now let's use the point-slope form of a line to find the equation of the line that passes through the point (-2, 6) with a slope of 5/4.
The point-slope form is given by:
y - y1 = m(x - x1)
Substituting the values into the equation:
y - 6 = (5/4)(x - (-2))
Simplifying:
y - 6 = (5/4)(x + 2)
Distributing (5/4) to (x + 2):
y - 6 = (5/4)x + (5/4)(2)
Simplifying:
y - 6 = (5/4)x + 10/4
Combining like terms:
y - 6 = (5/4)x + 5/2
To get the equation in standard form, let's move all the terms to one side:
(5/4)x - y = -5/2 + 6
Simplifying:
(5/4)x - y = -5/2 + 12/2
(5/4)x - y = 7/2
In conclusion, the equation of the line L that passes through the point (-2, 6) and is perpendicular to the line 4x + 5y = 10 is (5/4)x - y = 7/2.