How do you find a standard form equation for the line with point (2, 5) and is perpendicular to the line 4x + 9y − 7?

Question

Savannah Anderson · Accepted Answer

9x - 4y + 2 = 0

Given line is 4x + 9y - 7 = 0. or, 9y = - 4x + 7 or, y = - 4x/9 + 7/9 . It's slope is - 4/9. Let m be the slope of the perpendicular line. Then, m x ( - 4/9 ) = - 1 Hence m = 9/4 Let the required equation be y = 9/4 x + c. Since it passes through (2 , 5), we get c = 5 - 9/2 = 1/2. The equation is y = 9/4 x + 1/2 or, 4y = 9x + 2 or, 9x - 4y + 2 = 0

Kennedy Adams · Answer

undefined To find the standard form equation for a line perpendicular to $4x + 9y - 7$ and passing through the point $(2, 5)$, follow these steps:

1. Determine the slope of the given line by rearranging the equation into slope-intercept form ($y = mx + b$).
2. Find the negative reciprocal of the slope of the given line. This will be the slope of the perpendicular line.
3. Use the point-slope form ($y - y_1 = m(x - x_1)$) with the given point $(2, 5)$ and the slope found in step 2 to find the equation of the perpendicular line.
4. Convert the equation from point-slope form to standard form ($Ax + By = C$) by rearranging terms.

Applying these steps:

1. Given line: $4x + 9y - 7$
   - Rearrange into slope-intercept form: $9y = -4x + 7$
   - Divide by 9: $y = -\frac{4}{9}x + \frac{7}{9}$
   - Slope ($m$) of the given line: $-\frac{4}{9}$

2. Slope of perpendicular line: Negative reciprocal of $-\frac{4}{9}$ is $\frac{9}{4}$.

3. Use point-slope form with the point $(2, 5)$:
   $$y - 5 = \frac{9}{4}(x - 2)$$

4. Convert to standard form:
   $$4(y - 5) = 9(x - 2)$$
   $$4y - 20 = 9x - 18$$
   $$9x - 4y = -2$$

Therefore, the standard form equation for the line perpendicular to $4x + 9y - 7$ and passing through $(2, 5)$ is $9x - 4y = -2$.