Determine the appropriate description that matches the system of equations.
$\left\{\begin{array}{l}
y=5 x-15 \\
y=-5 x+15
\end{array}\right.$
a. Same line with infinite many solutions
b. Parallel lines with one solution
c. Parallel lines with no solution
d. Intersecting lines with one solution
Answer (1)
The system of equations is given by y = 5 x − 15 and y = − 5 x + 15 .
The slopes of the lines are 5 and − 5 , respectively, and the y-intercepts are − 15 and 15 , respectively.
Since the slopes are different, the lines intersect at one point.
Therefore, the system of equations represents intersecting lines with one solution: Intersecting lines with one solution .
Explanation
Analyze the equations We are given a system of two linear equations:
y = 5 x − 15
y = − 5 x + 15
We need to determine the relationship between these two lines and choose the correct description from the given options.
Identify slopes and intercepts Let's analyze the slopes and y-intercepts of the two lines.
The first line, y = 5 x − 15 , has a slope of 5 and a y-intercept of − 15 .
The second line, y = − 5 x + 15 , has a slope of − 5 and a y-intercept of 15 .
Compare slopes Since the slopes of the two lines are different ( 5 and − 5 ), the lines are not parallel and they are not the same line. Therefore, the lines must intersect at a single point.
Determine the relationship Intersecting lines have one solution, which corresponds to the point where the two lines meet.
Conclusion Therefore, the correct description for the given system of equations is:
Intersecting lines with one solution.
Examples
Understanding systems of equations is crucial in various real-life scenarios. For instance, consider a business trying to determine the break-even point where their revenue equals their costs. If the revenue and cost can be modeled as linear equations, finding the solution to the system of equations will give the production level needed to break even. Similarly, in physics, analyzing the motion of two objects often involves solving a system of equations that describe their positions over time. The intersection point represents when and where the objects meet, which is vital for collision detection and trajectory planning.
