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)
The equation − 9 ( x + 3 ) + 12 = − 3 ( 2 x + 5 ) − 3 x simplifies to − 15 = − 15 , which is always true. This indicates that there are infinitely many solutions. Therefore, the correct answer is option D.
;
Expand both sides of the equation: − 9 ( x + 3 ) + 12 = − 9 x − 27 + 12 = − 9 x − 15 and − 3 ( 2 x + 5 ) − 3 x = − 6 x − 15 − 3 x = − 9 x − 15 .
Simplify the equation: − 9 x − 15 = − 9 x − 15 .
Add 9 x to both sides: − 15 = − 15 .
Since the equation is always true, the equation has infinitely many solutions. D
Explanation
Analyze the problem We are given the equation − 9 ( x + 3 ) + 12 = − 3 ( 2 x + 5 ) − 3 x and asked to determine its solution type. This means we need to find out if it has one solution, no solution, or infinitely many solutions. We will start by expanding and simplifying both sides of the equation.
Expand the left side First, expand the left side of the equation: − 9 ( x + 3 ) + 12 = − 9 x − 27 + 12 = − 9 x − 15
Expand the right side Next, expand the right side of the equation: − 3 ( 2 x + 5 ) − 3 x = − 6 x − 15 − 3 x = − 9 x − 15
Simplified equation Now, we have the simplified equation: − 9 x − 15 = − 9 x − 15
Add 9x to both sides Add 9 x to both sides of the equation: − 9 x − 15 + 9 x = − 9 x − 15 + 9 x
− 15 = − 15
Conclusion Since − 15 = − 15 is always true, regardless of the value of x , the equation has infinitely many solutions.
Examples
Understanding equations with infinite solutions is crucial in various fields. For instance, in economics, when modeling supply and demand, an equation with infinite solutions might represent a market in perfect equilibrium, where any quantity can be sold at a specific price. Similarly, in physics, such equations can describe systems where energy is conserved, and multiple states satisfy the conservation law. Recognizing these scenarios helps in making informed decisions and predictions in real-world applications.
