Question

Create an inconsistent system of linear equations by graphing a line parallel to line w that passes through (-3,1)

Answer 1

how do i plot that?

Answer 2

To create an inconsistent system of linear equations, we need to understand that an inconsistent system occurs when the equations represent parallel lines, which do not intersect at any point.

1. **Define Line \( w \)**: First, we need the equation of line \( w \). For the purpose of this explanation, let's say line \( w \) has the equation:
\[
y = 2x + 4
\]

This line has a slope of \( 2 \).

2. **Determine a Parallel Line**: Lines are parallel if they have the same slope. Therefore, we will create a new line that has the same slope as line \( w \) (which is \( 2 \)) but passes through the point \((-3, 1)\).

We can use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is the point the line passes through and \( m \) is the slope. Plugging in our point \((-3, 1)\) and slope \( 2\):
\[
y - 1 = 2(x + 3)
\]
Simplifying this equation:
\[
y - 1 = 2x + 6
\]
\[
y = 2x + 7
\]

3. **Write the System of Equations**: We now have two equations that represent parallel lines:
- Line \( w \): \( y = 2x + 4 \)
- New parallel line: \( y = 2x + 7 \)

4. **Conclusion**: The system of equations:
\[
\begin{align*}
y &= 2x + 4 \quad \text{(Line w)} \\
y &= 2x + 7 \quad \text{(Parallel line through (-3, 1))}
\end{align*}
\]
is inconsistent because the two lines are parallel and do not intersect.

### Graphing
To visualize this:

- The line \( y = 2x + 4 \) crosses the y-axis at \( (0, 4) \) and has a slope of \( 2 \).
- The line \( y = 2x + 7 \) crosses the y-axis at \( (0, 7) \) and also has a slope of \( 2 \).

When graphed, you will see both lines running parallel to each other, confirming that they do not intersect, hence forming an inconsistent system.