6. The equation needs to be correct for every pair (x, y) of values from the table. If we want to find the right equation, we simply plug a pair from the table and see if it fits. Let's take the first pair (1, -2) and plug it in the equation a):
-2 = 1+2 ---> obviously not correct
Now let's do the same for equation b):
-2 = 3+3 ---> again incorrect
Equation c):
-2 =1-11 --> incorrect
And equation d):
-2 = 2-4 ---> correct
It's important to know that we could've chosen any pair from the table to check the equation. However, sometimes one pair can fit into more than one equation. In that case, we have to plug in other pairs as well to see which equation is the right one.
7. Scale factor, simply put, shows how many times one value of the figure changed when mapping onto another.
We are given the triangle ABC. Let's say the side of each of these little squares in 1 unit. That way, we conclude that the length AB is 4 units and BC is 3 units.
Now, let's look at the triangle A'B'C'. Length of the A'B' is 12 units and B'C' is 9 units.
Now to find the scale factor we simply have to find how many times are the sides of the mapped triangle greater then the triangle ABC:
AB/A'B' = BC/B'C' = 3
So, the scale factor is 3.
8. Note that the plumber charges two fees:
- base fee is a constant fee, for all appointments, regardless of the repair needed and hours spent.
- repair fee is a fee paid only if the repair is needed and is dependent on the number of hours spent repairing (more hours greater the fee)
So, in our equation
y = 25x + 30
y represents the total cost and x represents hours.
Now, we can see that this 25x part of equation represents repair fee, because it depends on x (for every hour spent, one pays another $25).
That means that 30 from our equation represents the base fee, the amount everyone will be charged regardless of the need for repair.
9. A relationship is linear if it's graph is a straight line, which means that its rate of change is constant. That means that the change in values of x of the two adjacent points, divided by the change of y is always the same number. That means that:
(x2 - x1)/(y2-y1) = (x3 - x2)/(y3 - y2) = (x4 - x3)/(y4 - y3) etc.
Let's test this.
Our first point is (0, 1) and next one is 1, 2). So, our first change of rate is:
(1-0)/(2-1) = 1/1 = 1
Let's find the rate of change for the next two adjacent pairs:
(x2, y2) = (1, 2)
(x3, y3) = (2, 4)
So, the rate of change is:
(2-1)/(4-2) = 1/2
It's obvious that the rate of change isn't always the same, which means this relation isn't linear.
