Solve the system of equations:
[tex]
\begin{array}{l}
x=2 y-5 \\
-3 x=-6 y+15
\end{array}
[/tex]
A. (1,-3)
B. (-2,-9)
C. No solution
D. Infinite solutions
Answer (2)
Rewrite the second equation and observe that it is identical to the first equation.
Recognize that the two equations represent the same line.
Conclude that the system has infinitely many solutions.
The solution is Infinite solutions .
Explanation
Analyze the problem We are given the following system of equations:
x = 2 y − 5
− 3 x = − 6 y + 15
We need to determine whether the system has a unique solution, no solution, or infinitely many solutions.
Simplify the equations Let's rewrite the second equation to see if it is related to the first equation. We have:
− 3 x = − 6 y + 15
Divide both sides by − 3 :
x = 2 y − 5
We observe that the second equation is just a multiple of the first equation.
Determine the solution Since both equations are identical, they represent the same line. Therefore, the system has infinitely many solutions. Any point ( x , y ) that satisfies the equation x = 2 y − 5 is a solution to the system.
Final Answer The system of equations has infinitely many solutions.
Examples
Systems of equations are used in various real-world applications, such as determining the optimal mix of products to maximize profit, balancing chemical equations, and modeling supply and demand in economics. In this case, recognizing that the two equations are dependent allows us to understand that there are infinite possible combinations that satisfy the given conditions. For example, if you're mixing two ingredients to achieve a certain outcome, and the ratio of the ingredients is fixed, you have infinite possible quantities that will work as long as the ratio is maintained.
The system of equations has infinitely many solutions because both equations simplify to the same expression, indicating they represent the same line. You can choose any value for y to find a corresponding x. Therefore, the answer is 'Infinite solutions'.
;
