A line is perpendicular to x+3y-4=0 and has the same y-intercept as 2x+5y-20=0. Find an equation for the line
y=-1/3x+4
To find the equation of a line perpendicular to the given line, we recall that the slopes of two perpendicular lines are negative reciprocals of each other.
The given line has the equation x + 3y - 4 = 0.
We can rewrite it in slope-intercept form (y = mx + b) by isolating y:
3y = -x + 4
y = (-1/3)x + 4/3
The slope of the given line is -1/3.
To find the slope of the perpendicular line, we take the negative reciprocal: -1/(-1/3) = 3.
We also know that the perpendicular line has the same y-intercept as the line with the equation 2x + 5y - 20 = 0.
To find the y-intercept of this line, we set x = 0 and solve for y:
2(0) + 5y - 20 = 0
5y = 20
y = 4
So, the y-intercept is 4.
We can now write the equation of the perpendicular line in slope-intercept form, using the slope (m = 3) and the y-intercept (b = 4):
y = mx + b
y = 3x + 4
Therefore, an equation for the line that is perpendicular to x+3y-4=0 and has the same y-intercept as 2x+5y-20=0 is y = 3x + 4.
To find an equation for the line that is perpendicular to x+3y-4=0, we first need to determine the slope of this line.
The given equation is in the standard form Ax + By + C = 0, where A, B, and C represent constants. The coefficient of the y-term, which is 3 in this case, can be considered as the slope.
Now, since we are looking for a line perpendicular to this, we need to find the negative reciprocal of the slope. The negative reciprocal of 3 is -1/3.
To find the equation of the perpendicular line, we can use the point-slope form, which is y - y1 = m(x - x1), where m represents the slope and (x1, y1) represents a point on the line.
The equation 2x+5y-20=0 is in the form Ax + By + C = 0. By rearranging the equation, we get it into the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. In this case, the y-intercept is -4.
So, we have the slope -1/3 and the y-intercept -4 for our desired line. Plugging these values into the point-slope form, we get:
y - (-4) = -1/3(x - x1)
Simplifying further:
y + 4 = -1/3(x - x1)
Therefore, an equation for the line that is perpendicular to x+3y-4=0 and has the same y-intercept as 2x+5y-20=0 is:
y + 4 = -1/3(x - x1)
Note: The equation is not fully determined as the point (x1, -4) on the line is not specified.
Here are two hints:
(1) The first equation tells you that the slope of the line you are looking for is m = 3. That is because the line you want to be perpendicular to has a slope of -1/3.
(2) The y-intercept of the line you want is b = 20/5 = 4. The point is located at (0,4)
y = mx + b
Substitute in the m and b values from above.