Question

The graph shows a line and two similar triangles.

On a coordinate plane, a line goes through (0, 0) and (6, 4). A small triangle has a rise of 2 and run of 3. A larger triangle has a rise of 4 and run of 6.

Which expression finds the equation of the line?
StartFraction y Over x EndFraction = three-halves
StartFraction y Over x EndFraction = two-thirds
StartFraction y Over 2 EndFraction = StartFraction 3 Over x EndFraction
StartFraction y Over 3 EndFraction = StartFraction x Over 2 EndFraction

Answer 1

To find the equation of the line that passes through the points (0, 0) and (6, 4), we can first determine the slope of the line.

The slope \( m \) is calculated as:

\[
m = \frac{\text{rise}}{\text{run}} = \frac{4 - 0}{6 - 0} = \frac{4}{6} = \frac{2}{3}
\]

This means that the equation of the line in slope-intercept form \( y = mx + b \) simplifies to:

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

Now we can rearrange this equation to find an equivalent expression. We start from:

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

Multiplying both sides of the equation by 3 gives:

\[
3y = 2x
\]

Now, dividing both sides by \( x \) (assuming \( x \neq 0 \)) gives:

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

Even though this representation isn’t in the original set of expression options, we can see that it aligns with the ratio found earlier.

Let's check the provided answer choices based on this calculation:

1. **StartFraction y Over x EndFraction = three-halves** → This implies \( \frac{y}{x} = \frac{3}{2} \) (not correct).
2. **StartFraction y Over x EndFraction = two-thirds** → This implies \( \frac{y}{x} = \frac{2}{3} \) (this is correct).
3. **StartFraction y Over 2 EndFraction = StartFraction 3 Over x EndFraction** → This doesn't equate correctly.
4. **StartFraction y Over 3 EndFraction = StartFraction x Over 2 EndFraction** → This doesn't equate correctly.

Given the analysis, the correct expression that finds the equation of the line is:

**StartFraction y Over x EndFraction = two-thirds**.