Question

ompare the equations −12x+y=8 and 12y=14x−4 . Identify the true statement.(1 point) Responses The equations have the same slope and y -intercept. The equations have the same slope and y -intercept. The equations have the same y -intercept, or initial value. The equations have the same y -intercept, or initial value. The equations have no common characteristics. The equations have no common characteristics. The equations have the same slopes and are parallel lines.

Answer 1

To compare the equations \(-12x + y = 8\) and \(12y = 14x - 4\), we will put both equations into the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

1. **Convert the first equation:**
\[
-12x + y = 8
\]
Add \(12x\) to both sides:
\[
y = 12x + 8
\]
Here, the slope \(m_1 = 12\) and the y-intercept \(b_1 = 8\).

2. **Convert the second equation:**
\[
12y = 14x - 4
\]
Divide every term by \(12\):
\[
y = \frac{14}{12}x - \frac{4}{12}
\]
Simplifying gives:
\[
y = \frac{7}{6}x - \frac{1}{3}
\]
Here, the slope \(m_2 = \frac{7}{6}\) and the y-intercept \(b_2 = -\frac{1}{3}\).

3. **Compare slopes and y-intercepts:**
- The slopes are \(m_1 = 12\) and \(m_2 = \frac{7}{6}\). Since \(12 \neq \frac{7}{6}\), they do not have the same slope.
- The y-intercepts are \(b_1 = 8\) and \(b_2 = -\frac{1}{3}\). Since \(8 \neq -\frac{1}{3}\), they do not have the same y-intercept.

Based on this analysis:

- The equations do not have the same slope or y-intercept.
- Therefore, the true statement is: **The equations have no common characteristics.**