Find the equation of a line which is
perpendicular to 2x + 3y = 4 passing through ( -5, 1)
Can you show mw how to work this out step by step please?
As an example:
3x + 2y = 5 passing through ( -3, 2)
The coordinates of the point is x1=-3, and y1=2.
The required equation is
2(x-x1)-3(y-y1)=0, i.e. ignore the constant term.
2(x-(-3))-3(y-2)=0
Notice how the coefficients of x and y have been switched around and the sign of one of the coefficients changed. This is done to get a line perpendicular to the original one.
Now simplify:
2(x-(-3))-3(y-2)=0
2x+6-3y+6=0
2x-3y=-12
Check by substituting x1 for x, and y1 for y:
2(-3)-3(2)=-12 OK
When the equations are in the normalized form : Ax+By+C=0
check for perpendicularity between 2x+3y=5 and 3x-2y=-12
the sum of the coefficients of x and the coefficients of y should be zero:
2*3+3*(-2)=0 the lines are perpendicular.
When the equations are of the form
y=mx+b and y=m1x+b1
then m*m1 should equal -1.
The problem with this form is that it does not work with vertical lines.
There should be enough information for you to work out your homework problem!
To find the equation of a line perpendicular to a given line, we first need to determine the slope of the given line.
Step 1: Convert the given equation into slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
Given equation: 2x + 3y = 4
Start by isolating y on one side of the equation.
3y = -2x + 4
Divide both sides of the equation by 3 to solve for y.
y = (-2/3)x + 4/3
Now we can determine the slope of the given line. The slope-intercept form tells us that the coefficient of x represents the slope. In this case, the slope (m) is -2/3.
Step 2: Determine the slope of the line perpendicular to the given line.
For a line to be perpendicular to another line, its slope must be the negative reciprocal of the slope of the given line.
To find the negative reciprocal, flip the fraction and change the sign.
The slope of the line perpendicular to the given line is 3/2 (negative reciprocal of -2/3).
Step 3: Use the point-slope form of a line to find the equation.
The point-slope form of a line is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
We are given a point on the line, (-5, 1), and the slope of the line perpendicular to the given line, which is 3/2.
Substituting the values into the point-slope form, we get:
y - 1 = (3/2)(x - (-5))
y - 1 = (3/2)(x + 5)
To convert this equation to slope-intercept form, simplify and solve for y:
y - 1 = (3/2)x + 15/2
y = (3/2)x + 15/2 + 2/2
y = (3/2)x + 17/2
Therefore, the equation of the line that is perpendicular to 2x + 3y = 4 and passes through the point (-5, 1) is y = (3/2)x + 17/2.