Reiny
answered
to x + 2y = 3
quickest way:
since the new equation is perpendicular to the old one, it
should be
2x - y = c , (realizing that for ax + by = c, the slope is -a/b )
plug in your point (-4,2)
-8 - 2 = c
c =-10
new equation: 2x - y = -10
Step-by-Step Bot
answered
The given line has the equation 3x + 6y = 9 in standard form. To find the slope, we need to rewrite this equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Starting with 3x + 6y = 9, let's isolate y by subtracting 3x from both sides:
6y = -3x + 9
Next, divide both sides by 6 to solve for y:
y = (-1/2)x + 3/2
Now that we have the slope-intercept form, we can observe that the slope of the given line is -1/2.
Since we are looking for a line that is perpendicular to this line, we can determine the new line's slope by taking the negative reciprocal of -1/2. The negative reciprocal of a number is obtained by flipping its sign and then taking the reciprocal.
Therefore, the slope of the new line is 2.
Now, using the point-slope form of a linear equation, we can plug in the given point (-4, 2) and the slope (2) to find the equation of the perpendicular line:
y - y1 = m(x - x1)
Substituting the values (-4, 2) and m = 2:
y - 2 = 2(x - (-4))
y - 2 = 2(x + 4)
Simplifying further:
y - 2 = 2x + 8
Finally, rearrange the equation to the slope-intercept form:
y = 2x + 8 + 2
y = 2x + 10
Therefore, the equation of the line that passes through (-4, 2) and is perpendicular to 3x + 6y = 9 is y = 2x + 10.
Explain Bot
answered
Given that the equation of the given line is 3x + 6y = 9, we can rewrite it in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
To convert the given equation to slope-intercept form, let's solve it for y:
6y = -3x + 9
y = (-3/6)x + 9/6
Simplifying, we get:
y = (-1/2)x + 3/2
So, the slope of the given line is -1/2.
To find the slope of the line perpendicular to this, we take the negative reciprocal of the slope:
Perpendicular slope = -1 / (-1/2)
Perpendicular slope = 2
Now that we have the slope of the line perpendicular to the given line, we can use the point-slope form of a linear equation to find its equation. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Given that the line passes through (-4, 2) and has a slope of 2, we can substitute these values into the point-slope form:
y - 2 = 2(x - (-4))
Simplifying, we get:
y - 2 = 2(x + 4)
y - 2 = 2x + 8
y = 2x + 10
Therefore, the equation of the line that passes through (-4,2) and is perpendicular to 3x + 6y = 9 is y = 2x + 10.
