Question

On a coordinate plane, a line goes through (0, 3) and (x, y). A triangle has a rise of 2 and run of 3. A larger triangle has a rise of 5 and run of 7.
Use similar triangles to determine the equation of the line with a slope of 2/3 that passes through the point (0, 3).

What is the ratio of the rise to the run in the smaller triangle in the diagram?
What is the ratio of the rise to the run in the larger triangle in the diagram?
What is the equation of the line in slope-intercept form?

Answer 1

To determine the equation of the line with a slope of \( \frac{2}{3} \) that passes through the point \( (0, 3) \), we can follow these steps:

### 1. Determine the ratios of rise to run for both triangles:

1. **Smaller Triangle**:
- Rise = 2
- Run = 3
- Ratio (Rise to Run) = \( \frac{\text{Rise}}{\text{Run}} = \frac{2}{3} \)

2. **Larger Triangle**:
- Rise = 5
- Run = 7
- Ratio (Rise to Run) = \( \frac{\text{Rise}}{\text{Run}} = \frac{5}{7} \)

### 2. Equation of the line in slope-intercept form:

The slope-intercept form of a line is given by:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.

Since we know the slope \( m = \frac{2}{3} \) and the y-intercept \( b = 3 \) (because the line passes through the point \( (0, 3) \)), we can substitute these values into the equation:

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

### Summary:

- **Ratio of rise to run in the smaller triangle**: \( \frac{2}{3} \)
- **Ratio of rise to run in the larger triangle**: \( \frac{5}{7} \)
- **Equation of the line in slope-intercept form**: \( y = \frac{2}{3}x + 3 \)