Question

Write the equation of the line perpendicular to 2x+3y=9 that passes through (-2,5). Write your answer in slope-intercept form. show work please.

Answer 1

To find the equation of a line perpendicular to 2x + 3y = 9 that passes through (-2, 5), we need to find the negative reciprocal of the slope of the given line and use the point-slope form of a linear equation.

Step-by-step explanation:

To find the equation of a line perpendicular to the given line, we need to find the negative reciprocal of the slope of the given line. The given line has the equation 2x + 3y = 9. To get it in slope-intercept form, we can solve for y:

3y = -2x + 9

y = (-2/3)x + 3

The slope of the given line is -2/3. The negative reciprocal of -2/3 is 3/2. So, the slope of the line perpendicular to the given line is 3/2. Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by:

y - y1 = m(x - x1)

Using the point (-2, 5) and the slope 3/2, we can substitute these values into the equation:

y - 5 = (3/2)(x - (-2))

y - 5 = (3/2)(x + 2)

y - 5 = (3/2)x + 3

y = (3/2)x + 8

Therefore, the equation of the line perpendicular to 2x + 3y = 9 and passing through (-2, 5) is y = (3/2)x + 8.

Answer 2

First find the slope of the first line. The line is in standard form, Ax + By = C, and slope is given by -A/B. Therefore, the slope of 2x+3y=9 is -2/3.

Now, if lines are perpendicular, their slopes are the negative reciprocals of each other. So, the slope of the line you want to find is the reciprocal of -2/3 (-3/2) multiplied by -1, or 3/2.

To find the equation of the line, we can use point-slope form because we have a point (-2, 5), and a slope that we calculated (3/2). Point-slope form is y-y₁=m(x-x₁) where (x₁, y₁) is a point and m is the slope. Our point is (-2, 5) so x₁ = -2 and y₁ = 5. Our slope is 3/2. Now, simply plug the values into the formula: y-5=3/2(x-(-2)). To convert the equation from point slope form to slope-intercept form, simply isolate y and get rid of parenthesis by distributing.