Question

The graphs of the lines represented by the equations
y=3x+7 and y=-3x-2

Answer 1

To analyze the graphs of the equations \( y = 3x + 7 \) and \( y = -3x - 2 \), we can start by determining their slopes and y-intercepts.

1. **Equation 1: \( y = 3x + 7 \)**
- Slope (\( m_1 \)): 3 (indicating the line rises steeply).
- Y-intercept: 7 (the line crosses the y-axis at (0, 7)).

2. **Equation 2: \( y = -3x - 2 \)**
- Slope (\( m_2 \)): -3 (indicating the line falls steeply).
- Y-intercept: -2 (the line crosses the y-axis at (0, -2)).

### Points of Intersection
To find the point of intersection of these two lines, set the equations equal to each other:

\[
3x + 7 = -3x - 2
\]

Now, solve for \( x \):

1. Add \( 3x \) to both sides:

\[
3x + 3x + 7 = -2
\]
\[
6x + 7 = -2
\]

2. Subtract 7 from both sides:

\[
6x = -2 - 7
\]
\[
6x = -9
\]

3. Divide by 6:

\[
x = -\frac{3}{2}
\]

Next, substitute \( x \) back into either original equation to find \( y \). Using the first equation:

\[
y = 3(-\frac{3}{2}) + 7
\]
\[
y = -\frac{9}{2} + \frac{14}{2}
\]
\[
y = \frac{5}{2}
\]

Thus, the point of intersection is

\[
\left(-\frac{3}{2}, \frac{5}{2}\right)
\]

### Graph Description
- The line \( y = 3x + 7 \) appears steep and rises from left to right, crossing the y-axis at (0, 7).
- The line \( y = -3x - 2 \) also appears steep but falls from left to right, crossing the y-axis at (0, -2).

The two lines intersect at the point \( \left(-\frac{3}{2}, \frac{5}{2}\right) \).

### Graphing the Lines
- To graph the lines accurately:
- For \( y = 3x + 7 \):
- When \( x = 0 \), \( y = 7 \).
- When \( x = -1 \), \( y = 3 \).
- When \( x = 1 \), \( y = 10 \).
- For \( y = -3x - 2 \):
- When \( x = 0 \), \( y = -2 \).
- When \( x = -1 \), \( y = 1 \).
- When \( x = 1 \), \( y = -5 \).

This gives a clear idea of how both lines would appear on a coordinate grid, showing their steep slopes and the point at which they intersect.