Select the correct answer.
Which statement is true about this equation?
[tex]$-9(x+3)+12=-3(2 x+5)-3 x$[/tex]
A. The equation has one solution, [tex]$x=1$[/tex].
B. The equation has one solution [tex]$x=0$[/tex].
C. The equation has no solution.
D. The equation has infinitely many solutions.
Answer (2)
Expand both sides of the equation: − 9 x − 27 + 12 = − 6 x − 15 − 3 x .
Simplify both sides: − 9 x − 15 = − 9 x − 15 .
Add 9 x to both sides: − 15 = − 15 .
Since the equation simplifies to a true statement, the equation has infinitely many solutions: D .
Explanation
Problem Analysis We are given the equation − 9 ( x + 3 ) + 12 = − 3 ( 2 x + 5 ) − 3 x and we need to determine if it has one solution, no solution, or infinitely many solutions.
Expanding Both Sides First, we expand both sides of the equation: − 9 x − 27 + 12 = − 6 x − 15 − 3 x
Simplifying Both Sides Next, we simplify both sides by combining like terms: − 9 x − 15 = − 9 x − 15
Adding 9x to Both Sides Now, we add 9 x to both sides of the equation: − 9 x − 15 + 9 x = − 9 x − 15 + 9 x − 15 = − 15
Conclusion Since the variables have been eliminated and we are left with a true statement, this means the equation has infinitely many solutions.
Examples
Understanding equations with infinite solutions is crucial in various fields. For instance, in electrical engineering, when designing circuits, certain configurations might lead to equations where the voltage or current can take infinitely many values under specific conditions. Similarly, in economics, models involving supply and demand might result in equations with infinite solutions, indicating market equilibrium over a range of prices. Recognizing these scenarios allows engineers and economists to make informed decisions and design robust systems.
The given equation simplifies to a true statement, indicating it has infinitely many solutions; thus, the correct answer is D.
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