Question

System of Equations:
p=6−2q
p=4+q
The values of p and q that solve these two equations simultaneously can be seen on the graph as (the coordinates at which the two lines intersect, the slopes of the two lines, the inverse of the slopes of the two lines, the values of the intercepts of the two lines).

Answer 1

To solve the system of equations given by:

1. \( p = 6 - 2q \)
2. \( p = 4 + q \)

we can find the point of intersection (the values of \( p \) and \( q \) that solve these equations simultaneously) by setting the two equations equal to each other:

\[
6 - 2q = 4 + q
\]

Now, we'll solve for \( q \):

1. Rearranging:
\[
6 - 4 = 2q + q
\]
\[
2 = 3q
\]
\[
q = \frac{2}{3}
\]

Next, we substitute the value of \( q \) back into either of the original equations to find \( p \). Let's use the second equation:

\[
p = 4 + q = 4 + \frac{2}{3}
\]
To add \( 4 \) and \( \frac{2}{3} \), we convert \( 4 \) to a fraction:
\[
4 = \frac{12}{3}
\]
So,
\[
p = \frac{12}{3} + \frac{2}{3} = \frac{14}{3}
\]

Thus, the solution to the system of equations is:
\[
(p, q) = \left(\frac{14}{3}, \frac{2}{3}\right)
\]

### Details about the Graph

#### Slopes of the Lines:
- For \( p = 6 - 2q \):
This can be rearranged to \( p = -2q + 6 \). The slope is \( -2 \).

- For \( p = 4 + q \):
This can be rearranged to \( p = q + 4 \). The slope is \( 1 \).

#### Inverse of the Slopes:
- The inverse of the slope of the first line (slope = -2) is:
\[
\text{Inverse of } -2 = -\frac{1}{2}
\]

- The inverse of the slope of the second line (slope = 1) is:
\[
\text{Inverse of } 1 = 1
\]

#### The Intercepts:
- For \( p = 6 - 2q \):
- When \( q = 0 \), \( p = 6 \) --> y-intercept: \((0, 6)\)
- When \( p = 0 \), \( 0 = 6 - 2q \Rightarrow 2q = 6 \Rightarrow q = 3\) --> x-intercept: \((3, 0)\)

- For \( p = 4 + q \):
- When \( q = 0 \), \( p = 4 \) --> y-intercept: \((0, 4)\)
- When \( p = 0 \), \( 0 = 4 + q \Rightarrow q = -4\) --> x-intercept: \((-4, 0)\)

### Summary:
- Point of Intersection: \(\left(\frac{14}{3}, \frac{2}{3}\right)\)
- Slopes: Line 1: \(-2\), Line 2: \(1\)
- Inverses of Slopes: Line 1: \(-\frac{1}{2}\), Line 2: \(1\)
- Intercepts:
- Line 1: y-intercept \((0, 6)\), x-intercept \((3, 0)\)
- Line 2: y-intercept \((0, 4)\), x-intercept \((-4, 0)\)