I have a task where I have to find the equation of ....

A line that passes through (0,7) at a gradient of 3.

* I Think this one is 3x+7*

A line that passes through (2,3) at a gradient of 2.

A line that passes through (4,3) at a gradient of -1.

A line that passes through (2,1) and (4,5)

A line that passes through (4,5) and (6,7)

A line that passes through (0,5) and (3,2).

2. You scared me with the word 'gradient' = number. Gradient is an advanced term in math for derivatives ...

In my language, I think you mean by gradient a slope for a linear function.

A linear function is a function y=a*x+b where y and x are variables. a,b are constants where a is the slope and b the place where this function crosses the y axes.

If you want to draw a line you only need to draw two (different) points and there is only one line that can be drawn which passes by both points.

Let the two points be (x1,y1) and (x2,y2). We find the slope first : slope = (y2-y1)/(x2-x1). For the next setp you just take one of the two points and continue the work.

That is why when we already give you the slope, we usually give you one points along with it.

We plug the coordinates of that point in the formula y=ax+b (where a=slope and we already have it) and isolate b.

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I do one exemple of each case :

A line that passes through (4,3) at a gradient of -1.

so y=ax+b.

so y=(-1)*x+b. We can write it y=-x+b. After, we take the point (4,3) where its x coordinate is 4 and the y coordinate is 3 and plug them in our formula.

3 = -(4) + b so b = 7. So our final answer is : y=-x+7=(-1)*x+7. (same answer)

A line that passes through (2,1) and (4,5)

so y=ax+b.

a=slope = (5-1)/(4-2) = 4/2=2. So we got the slope.

so y=2x+b

We take one of the two points. For example, (2,1) : 1=2(2) + b so 1=4+b do b=-3. So y=2x-3

(The answer would have been the same if we chose the point (4,5) because 5=2*4+b implies 5=8+b so b=-3. Same !)

3. Thank you.

Y=3X + 7
Y=2X - 1
Y=7 - X
Y=2X - 3
Y=3X - 11
Y=5 - X

4. Sorry : Actually you were right using the word gradient for slope when you have y as a function of 1 variable x.

-> A line that passes through (4,5) and (6,7)

slope = (7-5)/(6-4) = 2/2 = 1 so y=x+b

5=4+b so b=1

y=x+1.

When you have your answer, you can check if it is right or not by plugging in your two points one after the other. If your equation is good it should give you an equality in both cases :

point (4,5) : formula y=x+1 implies 5=4+1 which is right
point (6,7) : formula y=x+1 implies 7=6+1 which is also right.

If I take yours :

point (4,5) : formula y=3x-11 implies 5=3*4-11 which is wrong (it is enough to conclude that it is not the right formula)

For a 'yes' you must have both cases right and for a 'no' you just need one is wrong and it disqualifies the formula.