Question

A linear function has a slope of Negative StartFraction 7 Over 9 EndFraction and a y-intercept of 3. How does this function compare to the linear function that is represented by the equation y + 11 = Negative StartFraction 7 Over 9 EndFraction (x minus 18)?
It has the same slope and the same y-intercept.
It has the same slope and a different y-intercept.
It has the same y-intercept and a different slope.
It has a different slope and a different y-intercept.

Answer 1

Let's analyze the two linear functions given in the problem.

1. The first linear function is represented in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- For the first function, the slope is \(-\frac{7}{9}\) and the y-intercept is \(3\).
- Therefore, the equation of the first linear function can be written as:
\[
y = -\frac{7}{9}x + 3
\]

2. The second function is given by the equation:
\[
y + 11 = -\frac{7}{9}(x - 18)
\]
We need to put this in slope-intercept form. Let's first expand and rearrange it:
\[
y + 11 = -\frac{7}{9}x + \frac{7}{9} \cdot 18
\]
\[
y + 11 = -\frac{7}{9}x + 14
\]
Now isolate \(y\):
\[
y = -\frac{7}{9}x + 14 - 11
\]
\[
y = -\frac{7}{9}x + 3
\]

Now that we have both functions:
- The first function is \( y = -\frac{7}{9}x + 3 \).
- The second function is also \( y = -\frac{7}{9}x + 3 \).

From this, we can see that both functions have the same slope of \(-\frac{7}{9}\) and the same y-intercept of \(3\).

Therefore, the correct comparison is:
**It has the same slope and the same y-intercept.**