Question
Create an inconsistent system of linear equations by graphing a line parallel to line w that passes through (-3,1)
Answers
GPT-4o mini
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.
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.
how do i plot that?