Heather writes the equations below to represent two lines drawn on the coordinate plane.
[tex]\begin{aligned}
-6 x+18 y & =0 \\
4 x-12 y & =20
\end{aligned}[/tex]
After applying the linear combination method, Heather arrived at the equation [tex]$0=60$[/tex]. What conclusion can be drawn about the system of equations?
A. The equation has no solution; therefore, the system of equations has no solution.
B. The equation has a solution at [tex]$(0,60)$[/tex]; therefore, the system of equations has a solution at [tex]$(0,60)$[/tex].
C. The equation has infinite solutions; therefore, the system of equation as infinite solutions.
D. The equation has a solution at [tex]$(0,0)$[/tex]; therefore, the system of equations has a solution at [tex]$(0,0)$[/tex].
Answer (2)
The linear combination method leads to the equation 0 = 60 .
The equation 0 = 60 is a contradiction.
A contradiction implies that the system of equations has no solution.
Therefore, the system of equations has no solution: The equation has no solution; therefore, the system of equations has no solution.
Explanation
Understanding the Result The problem states that Heather used the linear combination method on the given system of equations and arrived at the equation 0 = 60 . We need to determine what this result implies about the system of equations.
Interpreting the Contradiction The equation 0 = 60 is a contradiction, meaning it is never true for any values of x and y . This indicates that the original system of equations has no solution.
Final Conclusion Therefore, the conclusion is that the equation has no solution, and consequently, the system of equations has no solution.
Examples
Imagine you're trying to solve a puzzle where two pieces are supposed to fit together, but no matter how you try, they just don't align. This is similar to a system of equations with no solution. In real life, this could represent a situation where you're trying to meet two conflicting requirements at the same time, such as trying to buy a house that's both affordable and in a very expensive neighborhood. The contradiction 0 = 60 shows that the requirements cannot be met simultaneously.
Heather's result of 0 = 60 indicates a contradiction, showing that there is no solution to the system of equations. This means the two lines represented by the equations do not intersect. Hence, the chosen multiple choice option is A: The equation has no solution; therefore, the system of equations has no solution.
;
