Give the slope-intercept form of the equation of the line that is perpendicular to 5x + 2y = 12 and contains the point (2, 3).

Question

Solve for y = mx + b in order to get the slope (m) and then substitute in the slope and the point to find b.

Answer 1

We need an equation in the form of

y = mx + b

First piece of information we have is "line that is perpendicular to 5x+2y = 12". So, what is the slope of this line? Let's write it in the slope/intercept form:

5x + 2y = 12 (Add -5x to both sides)

2y = 12 - 5x (divide both sides by 2)

y = 6 - 5/2 x = -5/2 x + 6

So, m (slope) is -5/2. The perpendicular line will have a slope of -1/m = -1 / (-5/2) = 2/5. Now we have

y = 2/5 x + b. So, what is b? We can use the second piece of information: "the line contains the point (2,3)" or for this line, when x = 2, y = 3. Let's substitute x = 2 and y = 3 in the equation and solve for b.

3 = (2/5)*2 + b

3 = 4/5 + b

b = 3 - 4/5 = 15/5 - 4/5 = 11/5

So, the line we are looking for is:

y = (2/5) x + 11/5

Answer 2

5x + 2y = 12

2y = -5x + 12

y = (-5/2)x + 6 ←This line has slope -5/2

Any line perpendicular to the given line has slope -1 / (-5/2) = 2/5

Equation of line with slope 2/5 and containing the point (2,3) is: y - 3 = (2/5)(x - 2)

y - 3 = (2/5)x - 4/5

y = (2/5)x + 11/5