Write the equation of the line that is perpendicular to 10x=-y and passes through point (4,2/3)
To find the equation of a line that is perpendicular to 10x=-y, we need to find its slope first.
10x = -y can be rewritten as y = -10x.
Comparing this equation to the slope-intercept form, y = mx + b, we can see that the slope (m) of the line is -10.
The slope of a line perpendicular to this line will be the negative reciprocal of -10.
The negative reciprocal of -10 is 1/10.
So, the slope of the line perpendicular to 10x = -y is 1/10.
Now, using the point-slope form of a line, we can write the equation:
y - y1 = m(x - x1)
where (x1, y1) is the given point (4, 2/3), and m is the slope (1/10).
Substituting the values, we have:
y - 2/3 = (1/10)(x - 4)
Multiplying through by 10 to eliminate the fraction:
10y - 20/3 = x - 4
Rearranging the equation:
x - 10y = -20/3 + 4
Simplifying:
x - 10y = -20/3 + 12/3
x - 10y = -8/3
Therefore, the equation of the line that is perpendicular to 10x = -y and passes through the point (4, 2/3) is x - 10y = -8/3.
To find the equation of a line that is perpendicular to the given equation, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
The given equation is 10x = -y. To rewrite this equation in the slope-intercept form (y = mx + b), let's solve for y by multiplying both sides of the equation by -1:
10x = -y
-10x = y
The equation is now y = -10x.
The slope of this line is -10. Since we want to find a line that is perpendicular to this one, we need to find the negative reciprocal of the slope. The negative reciprocal of -10 is 1/10.
Now, we have the slope (m = 1/10) and a point (4, 2/3) on the line.
Using the point-slope form of the equation of a line: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we can substitute the values into the equation:
y - 2/3 = (1/10)(x - 4)
To simplify, we can distribute 1/10 to the x - 4:
y - 2/3 = (1/10)x - 4/10
Combining the constant terms:
y - 2/3 = (1/10)x - 2/5
Now, if we rewrite this equation in the slope-intercept form (y = mx + b):
y = (1/10)x - 2/5 + 2/3
y = (1/10)x - 4/10 + 6/10
Simplifying the constants:
y = (1/10)x + 2/10
Finally, we can simplify the constants:
y = (1/10)x + 1/5
Therefore, the equation of the line that is perpendicular to 10x = -y and passes through the point (4, 2/3) is y = (1/10)x + 1/5.