Question

for some real number A, the graph of the line y=9a+1)x+8 in the standard (x,y) coordinate plane passes through (2,6). what is the slope of this line

Answer 1

To determine the slope of the line given by the equation \( y = (9a + 1)x + 8 \), we need to analyze the components of the equation.

In the slope-intercept form of a line \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, we can see that the slope \( m \) for our line is \( 9a + 1 \).

Next, we need to find the value of \( a \) such that the line passes through the point \( (2, 6) \). We can do this by substituting \( x = 2 \) and \( y = 6 \) into the line equation:

\[
6 = (9a + 1)(2) + 8
\]

Now, simplify the equation:

\[
6 = 2(9a + 1) + 8
\]
\[
6 = 18a + 2 + 8
\]
\[
6 = 18a + 10
\]

Now, isolate \( a \):

\[
6 - 10 = 18a
\]
\[
-4 = 18a
\]
\[
a = -\frac{4}{18} = -\frac{2}{9}
\]

With \( a \) determined, we can now substitute back to find the slope:

\[
m = 9a + 1
\]
\[
m = 9\left(-\frac{2}{9}\right) + 1
\]
\[
m = -2 + 1 = -1
\]

Therefore, the slope of the line is \(\boxed{-1}\).