m = (y2-y1) /(x2 – x1)

Given one point (x1,y1) on a line and the slope (m), you can figure out if a second point (x2, y2) is on that line by plugging all the known information in. Here, one point would be (4,1), and our slope is 7; so let’s put them in.

m = (y2 – y1) / (x2 – x1)

7 = (y2 – 1) / (x2 – 4)

We also know (half) of a second point (x, 8), and are told it is on the line. That means that by putting the second point in, we have an equation we can solve

7 = (8-1)/(x-4)

7 = 7 / (x-4)

Looking at it, I can tell that (x-4) will have to be one (1), but let’s go ahead and solve it mathematically by multiplying both sides of the equation by the quantity (x-4)

7(x-4) = 7(x-4)/(x-4)

Since any NON-ZERO number divided by itself is 1, and since 1 times any number is that number, we can drop that (x-4)/(x-4)

7(x-4) = 7

Then divide by 7

(x-4) 7/7 = 7/7

(x-4) = 1

and add 4

x-4 + 4 = 1 + 4

x = 5

y = mx + b

When I consider a line, sometimes I think of it starting at a particular place and ‘going’ to another particular place; alpha and omega, the beginning and the end, and the rest is in the middle, the going.

When I consider a line mathematically, in terms of a graph, I usually think of a piece of graph paper with one straight line on it. If I use the graph paper as a graph, I’d draw a horizontal line across the middle of the page, right on top of one of the pre-printed lines. I’d call this axis, the horizontal axis, the “X” axis, or technically, the abscissa. I’d draw a vertical line down the middle of the page for the ordinate, or Y axis (vertical axis). Now I have my 2 dimensional graph on which that straight line I mentioned is drawn.

For our purposes here, let’s say this straight line crosses the Y axis. Note that the line crosses the Y axis at one particular point and we call that point “b”. We’ll need this later.

If I were to arbitrarily pick one point on the line, I could draw a straight line from that point to the Y axis, and another to the X axis. I could draw many such lines, but I’m interested in only two, the two lines that are parallel and/or perpendicular, however you like to look at it, to the axes. You are already familiar with these 2 lines, because they are the lines which define the corresponding points on the axes.

Let’s say we had a piece of graph paper with the X and Y axes clearly marked. When I plot the point (2,4) on the graph, I actually start at the intersection of the two axes and follow along the X axis until I reach “2”, and then I trace up a perfectly vertical line until I meet the imaginary perfectly horizontal which crosses the Y axis at “4”. At the intersection of these two imaginary lines I’m tracing, that’s where I draw, or ‘plot’, my point, my (2,4). Yet, 2 over and 4 up.

Another way of describing this is:

1) Go to the graph and place your pencil on the intersection of the X and Y axes; the “origin”. [point (0,0)]

2) Draw a line, perfectly horizontal, along the X axis to the position “2” and stop. Obviously, the length of this line is 2 units. [point (2,0)]

3) From this point, draw a line, perfectly vertical, parallel the Y axis, 4 units long and stop.

4) Draw the point here. Thus plotting (2,4).

Looking at it this way, we can see that the line has an ‘x component’ and a ‘y component’, and we can say: “From any point on the line, go right (or left) ‘x’ units, then go up (or down) ‘y’ units, and you’ll end up back on the line”. This is true for any starting point on the line.

We can turn these two numbers into a ratio, say y/x. Sometimes this is called “the rise over the run”, or “the height over the distance”, “up and over”, etc. The important thing here is that for our line, every time our x component changes, so does our y component; and vice versa. And it does so in a specific way; a predictable way. “For every ‘x’ units, go ‘y’ units perpendicular.” And that ratio, y/x, that’s called the slope. The larger the ratio, the greater the value of y compared to x, the “steeper” the slope is.

Now we know 2 crucial things about our line. We know where it crosses the Y axis; remember “b” above? And we know its slope, y/x, called “m”. In fact, these two things are all we need to know about our line to know everything about it. By knowing the slope and the “Y intercept”, that is “m” and “b”, we can start from the point (0,b), the Y intercept, and we can locate, or define, or plot, every other point on the line.

And we do it by this equation: y = mx + b

So, what are you ‘given’ in the problem? “The slope is 7, and the givien [sic] coordinates are (x , 8) and (4 , 1)”. From these givens we know the slope, m, is 7; m = 7; and for one point (4,1) we know the point, namely x = 4 and y = 1. Plugging these values into our equation gives us 1=7*4 + b. Once you’ve simplified and found the value of “b”, you can use that in another equation: 8=7x+(the value of b), and solve for x.

(y2-y1)/(x2-x1)

so.. (1-8)/(4-x)

so you get (-7)/(4-x)=7 (which is the slope)

… simply solve for x by multiplying both sides by (4-X)

after simplifying a bit more you should get….

35=7x

therefore x =5