i need to see how to work a couple of these slope intercepts and parallel and perpendicular problems

Question

write the equation for the slope intercept

a is (14,-2) and b is (-3,15)

Answer 1

Hi Amber,

There are two main forms that you can write a line in:

Point-Slope: y-y₁ = m(x-x₁)
We use point-slope when we know the slope m and a point (x₁, y₁)

Slope-Intercept: y = mx + b
We use slope-intercept when we know the slope m and the y-intercept b

There is no fundamental difference between the lines produced by these two equations - in other words, we can convert between Slope-Intercept and Point-Slope form with a little bit of algebra.  

Sometimes, however, the problems don't give us all the bits we need for either form directly, but we have to figure them out.  That is the situation in this case, where we are given two points, but not the slope.

So the first step is to calculate the slope:

m = (y2 - y1) / (x2 - x1)
  = (15 - -2) / (-3 - 14)
  = 17 / -17
  = -1

Now, since we know the slope, and we ALSO know two different points, we can plug one of the points and the slope into the point-slope formula:

y-y1 =  m(x-x1)  Starting equation
y-15 = -1(x- -3) Plug in m=-1 and (x1,y1) = (-3,15)
y-15 = -1(x+3)   Minus a negative -> add a positive

But since the problem wants the answer in slope-intercept form, we must convert to this form using our basic algebra rules:

y-15    = -1(x + 3)    Starting equation
y-15    = -x - 3       Distribute -1 across (x + 3)
y-15+15 = -x - 3 + 15  Add 15 to both sides
y       = -x + 12      Combine terms, DONE

So the line, in slope-intercept form, is y = -x + 12.

Now you might ask why I plugged in (-3, 15) instead of (14, -2).  Well it doesn't actually matter - we could have plugged in either point and we would have gotten the same answer in the end.

 

Answer 2

Amber, we find the slope first when we are given two ordered pairs.  Using the slope formula, we have [15 - (-2)]/[(-3 - 14)] and this gives 17/-17 or -1 as the slope.  The slope formula requires us to subtract the y coordinates of the ordered pairs and this will be the numerator in the slope formula.  Then we subtract the x coordinates in the same order to find the denominator in the slope formula. 

Now we find our equation in slope-intercept form which is y = mx + b.   The m is slope and the b is the y-intercept.  We have y = -1x + b.  We can use either ordered pair to find b.  I will use (14, -2) and we have -2 = -1*14 + b.  This gives -2 = -14 + b and then we have 12 = b.  The slope-intercept equation is y = -1x + 12 or just y = -x + 12.

If we graphed this equation, the straight line would have slope -1 and it would cross the y-axis at 12.  Any line that is parallel to this line would have the same slope -1, but a different y-intercept.  Any line that is perpendicular to our line graphed by y = -x + 12 would have a slope that is the negative reciprocal of -1 which would be -(1/-1) or just 1.