Write the equation of the line through each point. Use slope-intercept form.

Question

through (3, 1) and perpendicular to -4x + y - 1 = 0

 

 

I don't know how to do this step by step.

Answer 1

Hi There:

 

Ok, so for this word problem, there are a few fundamentals to make sure you understand,  First, the slope (m) of any line is the distance in the vertical direction it "travels" per unit of distance it travels in the horizontal direction, represented as a fraction, simply, as:  "rise / run."  If the line passed through the zero point for each axis (0, 0), the the equation would be:

 

y = m(rise/run)*x.  

 

However, very often the line hits, or "intercepts" the vertical axis at a point different than 0 when it passes through 0 on the x access, a kind of vertical offset.  This is called the y-intercept, which we represent in the equation as "b," so:

 

y = mx + b

 

This is called slope-intercept form.  We need to know at least TWO points on the line in order to figure out the slope-intercept form for any line.  Also, for the purpose of your equation, you should know that any set of perpendicular lines will have negative slopes to another:  this is because they "travel" at 90 degrees from one another, so the slope of one is the negative of the slope of the other:  Imagine a line y = x, so it passes through points (-2, -2), (-1,-1), (0,0), (1,1), (2,2), (3,3), etc.  Its slope is rise (1) over run (1), so 1/1 = 1.  The line perpendicular to it will pass through (-2, 2), (-1, 1), (0,0) (1, -1), (2, -2), (3, -3), etc.  Its slope is rise (-1) over run (1), so -1.  But just because lines are perpendicular, though, doesn't mean they intersect at the same point on the graph.  Our example lines intersect at (0,0), but not all lines must pass through this point. 

 

So on to your equation.  We know one point from the given information (3,1).  Now, we need to find another point to calculate the slope.  You are given the equation for a perpendicular line, which we must put into slope -intercept form:

 

1)  -4x + y - 1 = 0

2)  y -1 = 4x

3)  y = 4x +1

 

Now that the perpendicular line is in the proper format, we can see that its slope is 4 (rise) / 1 (run).  Its slope is 4.

 

5)  For the line you are actually asked about, then, its slope must be -4, or -4 (rise) / 1 (run).

 

That gives us the "m" of the slope-intercept form.  The easiest way to do the next step is to "walk us back" to x = 0, which is the y-intercept.  So:

 

4)  If the line runs through (3, 1), then applying the slope, we know that the next point in the negative x direction is (2, 5).  That's because we subtracted -4 (rise) for the negative x direction--subtracting a negative is the same as adding.  Then, the points are (1, 9), and (0, 13).  Now, we have the y-intercept (b) at 13, and we have the slope -4(m), so the answer is:

 

y = -4x + 13

 

Done!  But, to be sure, we can check it against the point we are told in the given information is on this line (3, 1)

 

y = mx + b

(1) = -4(3) + 13

1 = -12 + 13

1 = 1

 

Works!  We got it.