write the equation of the line with x-intercept 6 that is perpendicular to 3x+5y=90
perpendicular lines have slopes that are negative reciprocals of each other
swap the coefficients, and change a sign ... 5x - 3y = 90
the x-intercept is 6 ... x equals 6 when y equals zero
... 5x - 3y = 30
To find the equation of the line that is perpendicular to the given line, we need to determine the slope of the given line first.
The given line has the equation 3x + 5y = 90. To find its slope, we need to rearrange the equation into slope-intercept form (y = mx + b), where m is the slope of the line.
Start by subtracting 3x from both sides of the equation:
5y = -3x + 90
Next, divide both sides of the equation by 5:
y = (-3/5)x + 18
The slope of the given line is -3/5.
Since the line we want to find is perpendicular to the given line, its slope will be the negative reciprocal of -3/5. The negative reciprocal of a given slope, m, can be found by flipping the fraction and changing its sign. In this case, the negative reciprocal is 5/3.
We also know that the line has an x-intercept of 6, which means it passes through the point (6, 0).
Using the point-slope form of a linear equation, which is y - y_1 = m(x - x_1), where (x_1, y_1) is a point on the line and m is the slope, we can substitute the values into the equation:
y - 0 = (5/3)(x - 6)
Simplifying, we get:
y = (5/3)(x - 6)
This is the equation of the line that is perpendicular to 3x + 5y = 90 and has an x-intercept of 6.