find the equation of a line that passes through the points (-1,1),(0,7),(1,9), and (2,13)
To find the equation of a line that passes through these points, we will use the point-slope form of a linear equation. The point-slope form equation is given by:
(y - y₁) = m(x - x₁)
where (x₁, y₁) represents a point on the line and m is the slope of the line.
Let's use the first two points (-1,1) and (0,7) to find the slope (m). The slope is given by:
m = (y₂ - y₁) / (x₂ - x₁)
Using the coordinates (-1,1) and (0,7):
m = (7 - 1) / (0 - (-1)) = 6 / 1 = 6
So, the slope (m) of the line is 6.
Now, let's choose one of the given points, say (0,7), and substitute it into our point-slope equation:
(y - 7) = 6(x - 0)
Simplifying, we get: y - 7 = 6x
To find the equation, we need to choose one more point from the given points and substitute it into our equation. Let's take (1,9):
(9 - 7) = 6(1 - 0)
2 = 6
This is not correct. Therefore, the equation is not defined by only two points of (-1,1) and (0,7).
To confirm our findings, let's try another combination of two points. Let's select (0,7) and (1,9):
m = (9 - 7) / (1 - 0) = 2 / 1 = 2
Using the point-slope form with (0,7):
(y - 7) = 2(x - 0)
Simplifying, we get: y - 7 = 2x
Now, substituting (1,9) into the equation:
(9 - 7) = 2(1 - 0)
2 = 2, which is correct.
Therefore, the equation of the line that passes through the points (-1,1), (0,7), (1,9), and (2,13) is:
y - 7 = 2x