How would I solve for a line equation if I know it's perpendicular to another line, and I have 1 point?

Question

Example: (this isn't the actual question, I just want to know how one would solve it, so that I can solve a similiar problem. If I just asked for the answer, that wouldn't help at all if I encountered another problem, eh?)

If y = ax + b is the equation of the line that passes through (6, 5) and is perpendicular to the line y = 4x + 3, find a and b

Answer 1

The line is perpendicular to the line y = 4x + 3, therefore the equation is

y = - 0.25 x + b

The line passes through (6, 5) therefore

5 = - 0.25 * 6 + b

b = 5 + 1.5 = 6.5

The equation of the line is

y = - 0.25 x + 6.5

Answer 2

You know that y=ax + b is perpendicular to the line y=4x + 3.
Perpendicular lines have slopes that are the negative reciprocals of each other, so in this case, the slope of y=ax+b would be -1/4 (since y=4x+3 has a slope of 4).

Now, you have y=-1/4*x + b. Since you know a point on this line (6,5), then you can plug in those values for x and y and solve for b:

5 = -1/4*6 + b
5 = -6/4 + b
5 + 6/4 = b
20/4 + 6/4 = b
26/4 = b
b = 6.5

So, the values for a and b are a = -0.25 and b = 6.5

Answer 3

To find the equation of any straight line, you always need to know 2 points on the line, or 1 point and its slope. This information is always given somewhere in the problem.
For your specific case here, you are told 1 point and its slope. Slope? Well,yes. If line A is perpendicular to line B, then the slope of line A is the negative reciprocal of line B. In math language, m of A = -1/m of B
Now, to your example.
The slope of y = 4x + 3 is 4
The equation you seek is thus y = -1/4x + b
(6,5) is a point on the line, so
5 = -1/4(6) + b, from which b = 5+ 6/4
b =20/4 + 6/4, b=26/4
Your equation is y=-1/4x+ 26/4

a= -1/4, b = 26/4

Answer 4

Well, when a line is perpindicular to another line, it means that it forms a 90 angle with it. In algebra all you have to do to reference this is to get the negative reciprocal of the original slope. On the other hand, the y-intercept will be a different value(b)

In this case: y=4x+3

the slope is 4...so the line perpindular to it will have a slope of -1/4

so, y=-1/4x +b

No sub in the point given and solve for b

5=(-1/4)(6)+b

b= 13/2

therefore, the line is y=-1/4x + 13/2

Answer 5

Use point slope form, not slope intercept form. As stated above, the slope of the perpendicular line is the negated reciprocal. 4 is the original slope, so the slope of the perpendicular line is -1/4. The line passes through (6,5), so you have everything you need for the equation in point slope form.

y-5=-1/4(x-6)

No need to solve for the y-intercept. If you want it in slope intercept form, then just solve that function for y.

y=-1/4x+26/4

or

y=-0.25x+6.5