Find the scalar equation of the line through the point (1, -4) and perpendicular to the line 2x + 5y – 3 = 0.
I assume that you know that the slope of 2x + 5y – 3 = 0 is -2/5 by just looking at it.
Then the slope of the new line must be 5/2
and its equation must be 5x - 2y + c = 0 . Plug in the given point to find c.
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 find the slope of the given line and then find the negative reciprocal of that slope. Let's break it down step by step:
Step 1: Convert the given equation to slope-intercept form (y = mx + b) by isolating y on one side:
2x + 5y – 3 = 0
5y = -2x + 3
y = (-2/5)x + 3/5
Step 2: Identify the slope (m) of the given line. In this case, the coefficient of x is -2/5, so the slope (m) is -2/5.
Step 3: Find the negative reciprocal of the slope. The negative reciprocal of a number is obtained by flipping the fraction and changing the sign. So, the negative reciprocal of -2/5 is 5/2.
Step 4: Now we have the slope (m) of our desired line, which is 5/2, and we also have a point (1, -4) that lies on the line.
Step 5: Use the point-slope form of the equation of a line to find the equation of the line with a given slope (m) and a point (x₁, y₁):
y - y₁ = m(x - x₁)
Substituting the values we have, we get:
y - (-4) = 5/2(x - 1)
y + 4 = 5/2(x - 1)
Step 6: Simplify the equation:
y + 4 = 5/2x - 5/2
y = 5/2x + 4 - 5/2
y = 5/2x + 3/2
Therefore, the scalar equation of the line through the point (1, -4) and perpendicular to the line 2x + 5y – 3 = 0 is y = 5/2x + 3/2.