Writing equations of perpendicular lines

we're asked what is the equation of line B and they tell us that line a has an equation y is equal to 2x plus 11 and they say that the line B contains the point 6 negative 7 and they tell us lines a and B are perpendicular perpendicular so that means that slope slope of B must be negative inverse of slope of a slope of a so what we'll do is we'll figure out the slope of a then take the negative inverse of it then we'll know the slope of B then we can use this point right here to fill in the gaps and figure out B's y-intercept so what's the slope of a this is already in slope-intercept form the slope of a is right there it's the two MX plus B so the slope here is equal to 2 so the slope of a is 2 what is the slope of B so what is B's B's slope going to have to be what's perpendicular a so it's going to be the negative inverse of this the inverse of 2 is 1/2 the negative inverse of that is negative 1/2 so B slope is negative 1/2 so we know that bees equation has to be y is equal to its slope M times X plus some y-intercept we still don't we still don't know what the y-intercept of B is but we can use this information to figure it out we know that Y is equal to negative 7 we know that Y is equal to negative 7 when X is equal to 6 when X is equal to 6 negative 1/2 times 6 times 6 plus B all right I'm just I just know that this is on the point so this this point must satisfy the equation of line B so let's work out what B must be so and this what or what the B the y-intercept is the lowercase B not the line B so we have negative 7 negative 7 is equal to what's negative 1/2 times 6 that's not a B there that's a six what's negative 1/2 times 6 it's negative three is equal to negative three plus our y-intercept let's add three to both sides of this equation so if we add 3 to both sides I just want to get rid of this 3 right here what do we get the left-hand side negative 7 plus 3 is negative 4 and that's going to be equal to these guys cancel out that's equal to be our y-intercept so this right here is a negative 4 so the equation of line B is is y is equal to its slope is a negative inverse of this character so negative 1/2 negative 1/2 X and its y-intercept we just figured out is negative 4 minus 4 and we are done