In the city of Marble Springs, Main Street intersects some of the other avenues in town. On a coordinate grid, Main Street passes through the points (-2,17) and (-22,33). Grimes Boulevard, a street that is perpendicular to Main Street, passes through the point (80,9). Write the equation of the line that represents Grimes Boulevard in slope intercept form. NO DECIMALS in your final answer and all fractions must be simplified.
slope of main street = (33-17)/(-22+2) = -4/5
so slope of Grimes Blvd = +5/4
y - 9 = (5/4)(x-80)
y = (5/4)x - 100+ 9
y = (5/4)x - 91
To find the equation of the line representing Grimes Boulevard in slope-intercept form, we first need to find the slope of the line.
The line that is perpendicular to Main Street will have a slope that is the negative reciprocal of the slope of Main Street.
The slope of Main Street can be calculated using the formula:
slope = (change in y)/(change in x)
Using the given points (-2,17) and (-22,33), the change in y is 33 - 17 = 16 and the change in x is -22 - (-2) = -20.
Therefore, the slope of Main Street is:
slope = (16)/(-20) = -4/5
The line perpendicular to Main Street will have a slope that is the negative reciprocal of -4/5, which is 5/4.
Now that we have the slope (m = 5/4) and a point on the line (80,9), we can use the point-slope form of a linear equation to find the equation of the line:
y - y1 = m(x - x1)
Substituting the values, we have:
y - 9 = (5/4)(x - 80)
To get the equation in slope-intercept form (y = mx + b), we need to simplify the equation and solve for y:
y - 9 = (5/4)x - (5/4)*80
y - 9 = (5/4)x - 400/4
y - 9 = (5/4)x - 100
y = (5/4)x - 100 + 9
y = (5/4)x - 91
Therefore, the equation of the line that represents Grimes Boulevard in slope-intercept form is y = (5/4)x - 91.