What a and b do to the line: slope, y-intercept, and parallel families

0:00Two constants in one rule, and each of them does exactly one job. y equals a x plus b. Change a and something happens to the line. Change b and something else happens. But change both at once and you learn nothing at all, because you cannot tell which of them caused what. So all of this is one method, run twice. Pin the constant at nothing and move the coefficient. Then pin the coefficient at two and move the constant.

0:34One experiment at a time. That is what makes the conclusions trustworthy rather than merely announced. Start with the constant pinned at nothing, so the rule is just y equals a x. Three of those. y equals half of x. y equals x. y equals two x. Drawn on one grid they fan out, and every one of them passes through the same point. That point is the origin, where the two axes cross. And it is not luck that they all go through it.

1:11Put nought in for x and watch. Whatever the coefficient happens to be, that coefficient times nought is nought, and with nothing added there is nothing else left. So the output is nought as well, and the pair nought nought satisfies every single rule of that shape. Which is a stronger statement than the drawing gave you. The drawing showed three lines. The substitution settles it for every coefficient nobody has drawn, the awkward ones included.

1:44One line of work, and the whole family is accounted for. Now steepness — and let us measure it rather than squint at it. Take one step to the right and see how far the line has climbed. On y equals half of x it climbs a half. On y equals x it climbs one. On y equals two x it climbs two. The rise for one step across is the coefficient itself. Not roughly: exactly, and by the same amount wherever along the line you choose to measure.

2:21So y equals x makes a natural landmark. It climbs one for one, leaning the same amount towards each axis. Above one is steeper than that; below one is gentler. The coefficient has a name. It is called the slope of the line. And the name earns its keep, because slope is now something you compute rather than an impression you form. Two lines, two coefficients, and the larger one is the steeper.

2:53For positive coefficients, at any rate. Which is where it starts to get interesting. You read the slope straight off the equation — provided the equation is in that shape to begin with. Two y equals four x plus seven is not. Divide the whole thing by two and it becomes y equals two x plus three and a half, so its slope is two. Three y equals six x minus eleven becomes y equals two x minus eleven thirds. Slope two again — and there was no way to see that before dividing through.

3:33Turn the sign over. y equals minus a third of x. y equals minus x. y equals minus three x. The same fan through the same origin, leaning the other way. Now a question. Which of those three is the steepest? Go by the number and minus three is the smallest of them, so minus three x ought to be the gentlest. Draw it, though, and it is plainly the sharpest of the three. So going by the number is simply wrong.

4:07Set the sign aside and look at the size instead. A third, one, three. Three is the largest, and minus three x is indeed the steepest. A third is the smallest, and minus a third of x is the shallowest. Measured the same way as before: one step across, and minus three x drops three while minus a third of x drops only a third. So the size of the coefficient decides how sharply the line leans, and the sign decides which way it leans. Two jobs from one number, and neither interferes with the other.

4:47Which means minus three x is exactly as steep as three x. Mirror images of each other. That sign carries a second reading, and it reaches back to something you have already met. A positive coefficient climbs. A negative one falls. And it is the coefficient that decides, never the constant. y equals two x minus fifty starts far below the axis and climbs the whole way regardless. Falling does not mean sitting below the axis either. y equals minus three x plus one falls steadily, and at every negative input it is above the axis.

5:32It crosses. Falling is a direction, not a position. Here is the connection worth the most, and it is the quietest one. Tiles in a growing pattern: one, then three, then five, then seven. The gap between consecutive stages is two, every time. Now set stage number against tile count and recover the rule. Stage one gives one and stage two gives three, so the rule is y equals two x minus one.

6:07Its coefficient is two. The gap was two. Those are not two facts that happen to agree. They are one number reached from opposite ends — counted down a table, and measured off a drawing. The steepness you see and the step you counted are the same quantity. Second experiment. Pin the coefficient at two and move the constant instead. y equals two x minus one. y equals two x plus one. y equals two x plus five.

6:42Watch what the change does, and more importantly what it does not do. The line does not tilt. It slides, bodily, with its direction locked. Which has to happen, because the rise for one step across is still two on all three of them. Moving the constant cannot touch it. And three lines going the same direction at different heights never meet. Not somewhere off the edge of the page. Nowhere at all.

7:13So where does a line meet the upright axis? That axis is exactly the points whose first coordinate is nought. So put nought in for x again. The coefficient's term vanishes, just as it did before, and what survives is the constant. The crossing sits at nought, b. y equals two x plus five crosses at nought, five. y equals x plus three crosses at nought, three. y equals three x minus two crosses at nought, minus two.

7:48That last one is worth pausing on. Minus two is below the origin. So the constant is a signed position on the axis rather than a distance. No distance is ever minus two. And there is the other half of the independence, sitting right there. Changing the coefficient does not move that crossing at all. y equals three x plus one and y equals minus three x plus one both cross at nought, one. One climbs and one falls, they are equally steep, and they meet the axis at exactly the same height.

8:29All of which gives you a test for parallel, and it is arithmetic rather than eyesight. Equal coefficients with different constants: parallel. Different coefficients: they cross somewhere, however alike they look. And they can look very alike. Three lines can lean the same general way, share no coefficient at all, and no two of them be parallel. Try a set. Three x minus one, three x, three x plus one. All three have coefficient three, so all three are parallel.

9:06Now try another. Minus two x minus three, minus two x, and two x plus three. The first two are parallel. The third has coefficient plus two rather than minus two, which makes it the mirror image, and it crosses both of the others. Compare the coefficients. Never the picture. And those two disguised rules from earlier both turned out to have coefficient two, so they are parallel — which nothing about the way they were written would have told you.