Find the standard form of the equation of the line that is perpendicular to -6x+5y=-6 and contains (8, -6)
By the way, thanks so much to anyone that can help me! (I forgot to add that before!)
The slope of the line
-6x+5y=-6
can be seen by rewriting it as
y = (6/5)x - 6/5
That slope is 6/5. A perpendicular line will therefore have slope -5/6.
Such a line passing through (8, -6) would have the equation
y +6 = (-5/6) (x-8)
y = -6 -(5/6)x + 40/6
y = (-5/6)x + (2/3)
To find the equation of a line that is perpendicular to the given line, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
Step 1: Convert the equation -6x + 5y = -6 to slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
-6x + 5y = -6
5y = 6x - 6
y = (6/5)x - 6/5
Step 2: Determine the slope of the given line. The equation is in the form y = mx + b, where m is the slope. Thus, the slope of the given line is 6/5.
Step 3: Find the negative reciprocal of the slope of the given line to get the slope of the perpendicular line. The negative reciprocal of 6/5 is -5/6.
Step 4: Use the point-slope form of a line, which is y - y₁ = m(x - x₁), where (x₁, y₁) represents the given point and m represents the slope of the line.
In this case, the given point is (8, -6) and the slope is -5/6. Plugging that into the equation, we get:
y - (-6) = (-5/6)(x - 8)
y + 6 = (-5/6)(x - 8)
Step 5: Simplify the equation.
y + 6 = (-5/6)x + 40/6
y + 6 = (-5/6)x + 20/3
Step 6: Convert the equation to standard form, which is Ax + By = C.
Multiply through by 6 to eliminate the fractions:
6(y + 6) = 6(-5/6)x + 6(20/3)
6y + 36 = -5x + 40
5x + 6y = 4
Therefore, the standard form of the equation of the line that is perpendicular to -6x + 5y = -6 and passes through (8, -6) is 5x + 6y = 4.