To determine how many solutions the given system of linear equations has, we need to analyze their slopes and intercepts. The system of equations provided is:
[tex]\[
\begin{array}{l}
y=\frac{4}{5} x -3 \\
y=\frac{4}{5} x + 1
\end{array}
\][/tex]
Let's break this down step-by-step:
1. Identify the slope and y-intercept of each equation:
- For the first equation [tex]\(y=\frac{4}{5} x - 3\)[/tex]:
- Slope ([tex]\(m_1\)[/tex]) = [tex]\(\frac{4}{5}\)[/tex]
- y-intercept ([tex]\(b_1\)[/tex]) = -3
- For the second equation [tex]\(y=\frac{4}{5} x + 1\)[/tex]:
- Slope ([tex]\(m_2\)[/tex]) = [tex]\(\frac{4}{5}\)[/tex]
- y-intercept ([tex]\(b_2\)[/tex]) = 1
2. Compare the slopes:
- The slope of the first equation is [tex]\(\frac{4}{5}\)[/tex].
- The slope of the second equation is also [tex]\(\frac{4}{5}\)[/tex].
- Since [tex]\(m_1 = m_2\)[/tex], the lines are parallel, indicating that they have the same direction.
3. Compare the y-intercepts:
- The y-intercept of the first equation is -3.
- The y-intercept of the second equation is 1.
- Since [tex]\(b_1 \neq b_2\)[/tex], the lines do not intersect at any point.
4. Determine the number of solutions:
- Since the slopes are equal and the y-intercepts are different, the lines are parallel and do not intersect.
- Parallel lines that do not intersect have no points in common.
- Therefore, this system of equations has no solutions.
In conclusion, the given system of linear equations has no solutions.
