Mr. Brown is creating examples of systems of equations. He completes the steps to find the solution of the equation below.
[tex]\begin{array}{c}
5 x+2 y=8 \
-4(1.25 x+0.5 y=2) \
\hline 5 x+2 y=8 \
\frac{-5 x-2 y=-8}{0=0}
\end{array}[/tex]
Based on this work, what is the solution to the system?
A. (-4,-4)
B. (0,0)
C. no solution
D. infinitely many solutions
Answer (2)
The system of equations is 5 x + 2 y = 8 and 1.25 x + 0.5 y = 2 .
Multiplying the second equation by -4 gives − 5 x − 2 y = − 8 .
Adding the equations results in 0 = 0 , indicating the equations are dependent.
Therefore, the system has infinitely many solutions .
Explanation
Analyze the System of Equations The given system of equations is:
5 x + 2 y = 8 1.25 x + 0.5 y = 2
Mr. Brown multiplied the second equation by -4, which resulted in:
− 4 ( 1.25 x + 0.5 y = 2 ) ⇒ − 5 x − 2 y = − 8
Then, he added the modified second equation to the first equation:
5 x + 2 y = 8 − 5 x − 2 y = − 8 0 = 0
The result 0 = 0 indicates that the two equations are linearly dependent, meaning they represent the same line. Therefore, there are infinitely many solutions to the system.
Determine the Solution Since the result of the addition is 0 = 0 , this means that the two equations are dependent and represent the same line. Therefore, any point on the line 5 x + 2 y = 8 is a solution to the system. This means there are infinitely many solutions.
Conclusion The system of equations has infinitely many solutions because the two equations are linearly dependent.
Examples
Consider a scenario where you're trying to determine the amount of ingredients for a recipe. If two different sets of instructions lead to the same ratio of ingredients, you have a system with infinitely many solutions. For example, if one recipe says you need 5 cups of flour and 2 cups of sugar, and another says you need 2.5 cups of flour and 1 cup of sugar, these are essentially the same recipe, just scaled differently. Any combination that maintains this ratio will work, illustrating infinitely many solutions.
The system of equations presented by Mr. Brown results in 0 = 0 , indicating that the equations are dependent and represent the same line. Therefore, the system has infinitely many solutions. The correct choice is D.
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