Let's go through each question one by one.
Question 1:
We have a point \((-12, -5)\) and a slope \(m = \frac{1}{3}\). We can use the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Plugging in the values:
\[ y - (-5) = \frac{1}{3}(x - (-12)) \] \[ y + 5 = \frac{1}{3}(x + 12) \] \[ y + 5 = \frac{1}{3}x + 4 \] \[ y = \frac{1}{3}x - 1 \]
So, the correct equation is: y = \frac{1}{3}x - 1
Question 2:
For a slope of 0 and a point \((3, -4)\), this means the line is horizontal. The equation for a horizontal line is of the form \(y = b\), where \(b\) is the y-value of the point:
\[ y = -4 \]
So, the correct equation is: y = -4
Question 3:
We have two points \((-4, 7)\) and \((6, 2)\). First, we need to find the slope \(m\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 7}{6 - (-4)} = \frac{-5}{10} = -\frac{1}{2} \]
Now we can use the point-slope form with one of the points, say \((-4, 7)\):
\[ y - 7 = -\frac{1}{2}(x + 4) \] \[ y - 7 = -\frac{1}{2}x - 2 \] \[ y = -\frac{1}{2}x + 5 \]
So, the correct equation is: y = -\frac{1}{2}x + 5
Question 4:
For the points \((-6, -3)\) and \((-4, -1)\), let's calculate the slope \(m\):
\[ m = \frac{-1 - (-3)}{-4 - (-6)} = \frac{2}{2} = 1 \]
Using the point-slope form with point \((-6, -3)\):
\[ y - (-3) = 1(x + 6) \] \[ y + 3 = x + 6 \] \[ y = x + 3 \]
So, the correct equation is: y = x + 3
Question 5:
From the pairs of \(x\) and \(y\) values, we can observe the points are: \((-4, 2)\), \((-2, 0)\), \((0, -2)\), \((2, -4)\), and \((4, -6)\).
Here, the change in \(y\) is -2 for every change of 2 in \(x\), indicating a slope of \(-1\):
Using point-slope form with one of the points, say \((0, -2)\):
\[ y - (-2) = -1(x - 0) \] \[ y + 2 = -x \] \[ y = -x - 2 \]
So, the correct equation is: y = -x - 2
Summary of Answers:
- y = \(\frac{1}{3}x - 1\)
- y = -4
- y = -\(\frac{1}{2}x + 5\)
- y = x + 3
- y = -x - 2