Which of the following is a point-slope equation of a line that passes through the points (-1,4) and (8,-2)
1 answer:
Answer:
y = (2/3)x + 4 2/3
Step-by-step explanation:
For reference, a linear function has an equation of y = mx + b
The first thing we're going to do is use the slope formula, or rise over run. In this case that looks like:
(-2-4) - (8 - -1)
-2 -4 comes out to - 6, and the denominator comes out to 9.
6/9 is equal to 2/3, which is our m or slope value.
Next we want to find the b value, or our y-intercept. To do that we'll plug one of the points into the equation with our new-found slope:
4 = (2/3)-1 +b
4 = -2/3 + b
4 2/3 = b
That means our complete equation is:
y = (2/3)x + 4 2/3
OR
y = (2/3)x + 14/3 (equivalent but an improper fraction rather than a mixed fraction).
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The pattern is that the numbers in the right-most and left-most squares of the diamond add to the bottom square and multiply to reach the number in the top square.
For example, in the first given example, we see that the numbers 5 and 2 add to the number 7 in the bottom square and multiply to the number 10 in the top square.
Another example is how the numbers 2 and 3 in the left-most and right-most squares add up to the number 5 in the bottom square and multiply to the number 6 in the top square.
Using this information, we can solve the five problems on the bottom of the paper.
a) We are given the numbers 3 and 4 in the left-most and right-most squares. We must figure out what they add to and what they multiply to:
3 + 4 = 7
3 x 4 = 12
Using this, we can fill in the top square with the number 12 and the bottom square with the number 7.
b) We are given the numbers -2 and -3 in the left-most and right-most squares, which again means that we must figure out what the numbers add and multiply to.
(-2) + (-3) = -5
(-2) x (-3) = 6
Using this, we can fill the top square in with the number 6 and the bottom square with the number -5.
c) This time, we are given the numbers which we typically find by adding and multiplying. We will have to use trial and error to find the numbers in the left-most and right-most squares.
We know that 12 has the positive factors of (1, 12), (2,6), and (3,4). Using trial and error we can figure out that 3 and 4 are the numbers that go in the left-most and right-most squares.
d) This time, we are given the number we find by multiplying and a number in the right-most square. First, we can find the number in the left-most square, which we will call . We know that
, so we can find that
, or the number in the left-most square, is 8. Now we can find the bottom square, which is the sum of the two numbers in the left-most and right-most squares. This would be
. The number in the bottom square is
.
e) Similar to problem c, we are given the numbers in the top and bottom squares. We know that the positive factors of 8 are (1, 8) and (2, 4). However, none of these numbers add to -6, which means we must explore the negative factors of 8, which are (-1, -8), and (-2, -4). We can see that -2 and -4 add to -6. The numbers in the left-most and right-most squares are -2 and -4.