Question

Graph the line that passes through the given point and has the given slope m.
1.) (-1,-3); m= 2
2.) (-3,-2);m=-4

Answer 1

To graph a line given a point and a slope, we can use the point-slope form of a linear equation, which is given by:

\[
y - y_1 = m(x - x_1)
\]

where \((x_1, y_1)\) is the given point and \(m\) is the slope.

### 1.) For the point \((-1,-3)\) with \(m = 2\):

Using the point-slope form:

\[
y - (-3) = 2(x - (-1))
\]

This simplifies to:

\[
y + 3 = 2(x + 1)
\]

Expanding it:

\[
y + 3 = 2x + 2 \\
y = 2x + 2 - 3 \\
y = 2x - 1
\]

Now, we can graph the line \(y = 2x - 1\).

- **Point:** Start at the point \((-1, -3)\) on the graph.
- **Slope:** The slope \(m = 2\) means that for every 1 unit you move to the right (positive x-direction), you will move up 2 units (positive y-direction).

Using the point \((-1, -3)\):
- Move to \((0, -1)\): 1 unit right, 2 units up from the point \((-1, -3)\).
- Another point: From \((0, -1)\) move to \((1, 1)\): again, 1 unit right, 2 units up.

Plot those points on the graph and draw a line through them.

### 2.) For the point \((-3,-2)\) with \(m = -4\):

Using the point-slope form:

\[
y - (-2) = -4(x - (-3))
\]

This simplifies to:

\[
y + 2 = -4(x + 3)
\]

Expanding it:

\[
y + 2 = -4x - 12 \\
y = -4x - 12 - 2 \\
y = -4x - 14
\]

Now, we can graph the line \(y = -4x - 14\).

- **Point:** Start at the point \((-3, -2)\) on the graph.
- **Slope:** The slope \(m = -4\) means that for every 1 unit you move to the right (positive x-direction), you move down 4 units (negative y-direction).

Using the point \((-3, -2)\):
- Move to \((-2, -6)\): 1 unit right, 4 units down from the point \((-3, -2)\).
- Another point: From \((-2, -6)\) move to \((-1, -10)\): again, 1 unit right, 4 units down.

Plot those points on the graph and draw a line through them.

### Summary:
1. For \((-1, -3)\) and \(m = 2\), the line is \(y = 2x - 1\).
2. For \((-3, -2)\) and \(m = -4\), the line is \(y = -4x - 14\).