Find the scalar equation of the line through the point (1, -4) and perpendicular to the line 2x + 5y – 3 = 0.
To find the scalar equation of the line through the point (1, -4) and perpendicular to the line 2x + 5y - 3 = 0, follow these steps:
Step 1: Write the given equation in slope-intercept form.
2x + 5y - 3 = 0
5y = -2x + 3
y = (-2x + 3) / 5
Step 2: Take the negative reciprocal of the slope of the given line to find the slope of the perpendicular line.
The slope of the given line is -2/5, so the slope of the perpendicular line is 5/2.
Step 3: Use the point-slope form to write the equation of the perpendicular line.
y - y1 = m(x - x1)
Using the point (1, -4) and the slope 5/2, we have:
y - (-4) = (5/2)(x - 1)
y + 4 = (5/2)x - 5/2
y = (5/2)x - 5/2 - 4
y = (5/2)x - 13/2
Step 4: Write the equation in scalar form by multiplying both sides by 2 to remove fractions.
2y = 5x - 13
The scalar equation of the line through the point (1, -4) and perpendicular to the line 2x + 5y - 3 = 0 is 2y = 5x - 13.
To find the scalar equation of the line through the point (1, -4) and perpendicular to the line 2x + 5y - 3 = 0, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
First, let's rewrite the equation of the given line in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
2x + 5y - 3 = 0
5y = -2x + 3
y = (-2/5)x + 3/5
From this equation, we can see that the slope of the given line is -2/5.
Now, since the line we're looking for is perpendicular to the given line, its slope will be the negative reciprocal of -2/5. To find the negative reciprocal, we flip the fraction and change its sign:
Negative reciprocal of -2/5 = 5/2
So, the slope of the line we're looking for is 5/2.
Next, we can use the point-slope form of a line to write the equation. The point-slope form is given by: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Plugging in the coordinates of the given point (1, -4) and the slope 5/2, we have:
y - (-4) = (5/2)(x - 1)
y + 4 = (5/2)(x - 1)
To get the scalar equation, we can distribute 5/2 to the terms in brackets:
y + 4 = (5/2)x - 5/2
y = (5/2)x - 5/2 - 4
y = (5/2)x - 13/2
So, the scalar equation of the line through the point (1, -4) and perpendicular to the line 2x + 5y - 3 = 0 is y = (5/2)x - 13/2.
using the point-slope form,
y+4 = 5/2 (x-1)