Question

write an equation of the line

1 |
|
2 |
__________| ______________
-2 |
|
-1 |

Answer 1

To determine the equation of a line based on the points on the provided graph, we first need to identify two points that the line passes through. Given the graph coordinates you've provided, let’s assume they represent the following two points:

1. Point A: (-2, 2)
2. Point B: (1, 1)

We can find the equation of the line that passes through these two points using the slope-intercept form of a line, \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

### Step 1: Calculate the slope \(m\)

The slope \(m\) is calculated using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using points A and B:

- \( (x_1, y_1) = (-2, 2) \)
- \( (x_2, y_2) = (1, 1) \)

Substituting the values:

\[
m = \frac{1 - 2}{1 - (-2)} = \frac{-1}{3}
\]

### Step 2: Use the slope and one point to find the y-intercept \(b\)

We can now use one of the points (we'll use point A: (-2, 2)) to find \(b\):

Using the point-slope form:

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

Substituting \(m\), \(x_1\), and \(y_1\):

\[
y - 2 = -\frac{1}{3}(x + 2)
\]

Distributing the slope:

\[
y - 2 = -\frac{1}{3}x - \frac{2}{3}
\]

Now, add 2 to both sides:

\[
y = -\frac{1}{3}x - \frac{2}{3} + 2
\]

Convert 2 to a fraction with a denominator of 3:

\[
y = -\frac{1}{3}x - \frac{2}{3} + \frac{6}{3}
\]
\[
y = -\frac{1}{3}x + \frac{4}{3}
\]

### Final Equation

Therefore, the equation of the line in slope-intercept form is:

\[
y = -\frac{1}{3}x + \frac{4}{3}
\]

If you prefer the equation in standard form, you can multiply through by 3 to eliminate the fraction:

\[
3y = -x + 4 \quad \Rightarrow \quad x + 3y = 4
\]

### Standard Form

The line can also be represented in standard form as:

\[
x + 3y = 4
\]