To create a system of linear equations consisting of the line \( w \) and a line perpendicular to it that passes through the point \( (1, -8) \), we need to follow these steps:
Step 1: Determine the slope of line \( w \)
The equation of line \( w \) is given as:
\[
y = -\frac{2}{5}x + 4
\]
From this equation, we can see that the slope \( m_w \) of line \( w \) is \( -\frac{2}{5} \).
Step 2: Find the slope of the perpendicular line
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, we calculate the slope \( m_p \) of the perpendicular line:
\[
m_p = -\frac{1}{m_w} = -\frac{1}{-\frac{2}{5}} = \frac{5}{2}
\]
Step 3: Write the equation of the perpendicular line
We know the slope \( m_p = \frac{5}{2} \) and it passes through the point \( (1, -8) \). We'll use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Substituting in the values:
\[
y - (-8) = \frac{5}{2}(x - 1)
\]
This simplifies to:
\[
y + 8 = \frac{5}{2}(x - 1)
\]
Now, let's simplify:
-
Distribute the slope:
\[
y + 8 = \frac{5}{2}x - \frac{5}{2}
\]
-
Subtract \( 8 \) from both sides to isolate \( y \):
\[
y = \frac{5}{2}x - \frac{5}{2} - 8
\]
-
Convert \( 8 \) to a fraction with a denominator of 2:
\[
8 = \frac{16}{2}
\]
So:
\[
y = \frac{5}{2}x - \frac{5}{2} - \frac{16}{2} = \frac{5}{2}x - \frac{21}{2}
\]
Final equations
The consistent system of linear equations consisting of line \( w \) and the newly formed perpendicular line is:
- \( y = -\frac{2}{5}x + 4 \) (line \( w \))
- \( y = \frac{5}{2}x - \frac{21}{2} \) (perpendicular line)
Graphing
If you were to graph these two equations:
- Line \( w \) has a negative slope and crosses the y-axis at \( 4 \).
- The perpendicular line would rise steeply, with a slope of \( \frac{5}{2} \), and it would cross the y-axis at \( -\frac{21}{2} \) or \( -10.5 \).
This system of equations will be consistent, as they will intersect at a point.
