Maths Homework - straight line equations

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My ds has this homework for Monday. His notes jotter has so many conflicting ways to work things out! We're really unsure about how go about some questions. We're stuck on questions 3, 4 and 5 but have managed to do the other ones. Any help would be appreciated.

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Answers here.

You need to rearrange the equations in q3 and q4 so that they are of the format:
nx + N = y (n and N are numbers). When y is on its own then n (the number x is multiplied by) is the gradient of the line, and N (the single number) is the y intercept, ie the place where the line crosses the y axis.

You need to rearrange equations so that they are in the form y = mx + b.

So for something like 4x - 2y + 6 = 0, move the ys over to the other side, to get 4x + 6 = 2y. Now divide everything by 2, so that you only have one y.
2x + 3 = y, or if you flip it round, y = 2x + 3.

Once it's in that form, the number before the x is the gradient (in this case, 2), and the number at the end (in this case +3) in the y-intercept.

Parallel lines have the same gradient, so in this case, you need another line with gradient of 2. Any line that has a 2 in front of the x in that format will be parallel, but they want a specific line that goes through the point 4, 6, so you need to find the y-intercept of that line (as that is what distinguishes one parallel line from another).

Start your equation y = 2x + b (because 'b' is the intercept, and what you don't know). You do know that when x = 4, y = 6, because that's the point they've given you, so substitute those in.

6 = 2(4) + b
6 = 8 + b
-2 = b

(using algebra to solve for 'b').

Now you know everything you need for the equation: y = 2x -2

That's for number 3.

For number 4, you only need the gradient and intercept, so just re-arrange the equation: 3y -2x = 6. Add 2x to each side,
3y = 2x + 6
then divide by 3, so that you only have one y.
y = 2/3 x + 2

So the gradient is 2/3 and the intercept is +2.

(I'm going back and forth between my post and your photo, so it's possible I've copied some of the questions wrong, but it should give you the basic idea of how to do it, anyway).

For the last one, you need to look at how much y changes compared to x. You can draw a triangle to help, where it crosses easy-to-see points. You can see that y goes down 3 squares in the space that x goes across 2. That means the gradient is 3/2 (always y on top, x on the bottom), but it has to be negative, as when x increases by 2, y is going down by 3 (or vice versa).

you can see from the graph that the intercept is +2, so the equation is

y = -3/2x +2

But you should write P and Q really, as that's in the graph, so

Q = -3/2 (P) + 2.

When Q = -7, you substitute that in to find P.

-7 = -3/2 (P) + 2
-9 = -3/2(P) (from -2 from both sides).
-18 = -3(P) (after multiplying both sides by 2).
6 = P (dividing by -3).

So when P = 6, Q = -7

His jotter might show different ways to work things out, and that's OK. They aren't really conflicting in the end (though they might appear so). It's just that there might be several ways to approach it, and some are easier to do in your head than others, or some are easier to do if you have graph paper and can see the graph, or others are easier if you are good at algebra or fractions, etc.

e.g., finding the gradient - you can either think to yourself: "if x goes one square to the right, what does y do?" and the answer to that will be the gradient. But, it can be hard to tell sometimes, if it cuts somewhere through the middle of the square. So on ones like that, you might look for points where the line cuts through a place where both x and y are whole numbers. Then you can say "when x goes 3 whole spaces to the right (for example), y goes up 5 whole squares". That tells you that the gradient is 5/3. It is the same answer that you'd get by asking the first way 'how much does y change when x moves one to the right', but seeing that y goes up 5/3 squares (2 and 2/3) might be kind of hard to tell. So it was easier to find ones that just used exact integers. It doesn't mean either way was more correct, though.

Note also that there is a picky difference depending on how questions are worded:

if it asks for the y-intercept, the answer should be a number, like "2".

If it asks for the co-ordinates of the y-intercept, or the co-ordinates of where the line crosses the y-axis, or some other wording like that, then the answer can be a pair of numbers as a co-ordinate, like (0,2). But some teachers will mark (0,2) wrong, if the question just asks for the intercept itself, which is 2.