Question

Do the equations y+6.5=3(x+1.5) and y−11.5=2(x+3.5) represent the same line?

Yes, because they both convert to y=2x+18.5 in slope-intercept form.

Yes, because they both convert to y=3x−2 in slope-intercept form.

Yes, because they both convert to y=3x+8 in slope-intercept form.

No. They cannot possibly represent the same line because they have different slopes.

Answer 1

To determine if the equations \( y+6.5=3(x+1.5) \) and \( y−11.5=2(x+3.5) \) represent the same line, we need to convert both equations to slope-intercept form (\( y = mx + b \)) and then compare their slopes and y-intercepts.

1. **For the first equation:**
\[
y + 6.5 = 3(x + 1.5)
\]
Distributing the \( 3 \):
\[
y + 6.5 = 3x + 4.5
\]
Subtracting \( 6.5 \) from both sides:
\[
y = 3x + 4.5 - 6.5
\]
\[
y = 3x - 2
\]

2. **For the second equation:**
\[
y - 11.5 = 2(x + 3.5)
\]
Distributing the \( 2 \):
\[
y - 11.5 = 2x + 7
\]
Adding \( 11.5 \) to both sides:
\[
y = 2x + 7 + 11.5
\]
\[
y = 2x + 18.5
\]

Now we can compare the two slope-intercept forms we found:

- The first equation converts to \( y = 3x - 2 \).
- The second equation converts to \( y = 2x + 18.5 \).

Since the slopes are different (3 versus 2), the two equations represent different lines.

Thus, the correct answer is:
**No. They cannot possibly represent the same line because they have different slopes.**