How many intersections are there of the graphs of the equations below?
[tex]\begin{array}{l}
\frac{1}{2} x+5 y=6 \\
3 x+30 y=36
\end{array}[/tex]
A. none
B. one
C. two
D. infinitely many
Answer (2)
Multiply the first equation by 6 to obtain an equivalent equation: 3 x + 30 y = 36 .
Compare the modified first equation with the second equation: 3 x + 30 y = 36 .
Observe that the two equations are identical, meaning they represent the same line.
Conclude that there are infinitely many intersections: in f ini t e l y man y .
Explanation
Understanding the Problem We are given two linear equations: 2 1 x + 5 y = 6 3 x + 30 y = 36 We want to find the number of intersections of the graphs of these equations. The number of intersections corresponds to the number of solutions to the system of equations.
Simplifying the First Equation To determine the number of intersections, we can manipulate the first equation to see if it is equivalent to the second equation. Multiply the first equation by 6: 6 ( 2 1 x + 5 y ) = 6 ( 6 ) This simplifies to: 3 x + 30 y = 36
Comparing the Equations Now, we compare the simplified first equation with the second equation: 3 x + 30 y = 36 3 x + 30 y = 36 Since the two equations are identical, they represent the same line.
Determining the Number of Intersections Since the two equations represent the same line, there are infinitely many points of intersection.
Final Answer Therefore, the graphs of the two equations have infinitely many intersections.
Examples
Understanding intersections of lines is crucial in various fields. For instance, in economics, the intersection of supply and demand curves determines the market equilibrium point. In navigation, the intersection of lines of position can pinpoint a vessel's location. Moreover, in computer graphics and game development, detecting intersections between objects is essential for collision detection and realistic interactions. This concept extends to more complex scenarios, such as network analysis and resource allocation, where understanding intersections helps optimize efficiency and avoid conflicts.
The graphs of the two equations intersect at infinitely many points because they represent the same line. After simplifying the first equation, both equations become identical. Thus, the answer is D. infinitely many.
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