How many solutions exist for the given equation?
[tex]3(x+10)+6=3(x+12)[/tex]
A. zero
B. one
C. two
D. infinitely many
Answer (1)
Distribute: 3 x + 30 + 6 = 3 x + 36
Combine like terms: 3 x + 36 = 3 x + 36
Subtract 3 x from both sides: 36 = 36
The equation simplifies to a true statement, so there are in f ini t e l y man y solutions.
Explanation
Analyze the equation We are given the equation 3 ( x + 10 ) + 6 = 3 ( x + 12 ) and asked to find the number of solutions. To do this, we will expand and simplify the equation.
Distribute First, distribute the 3 on both sides of the equation: 3 x + 30 + 6 = 3 x + 36
Combine Like Terms Next, combine like terms on both sides: 3 x + 36 = 3 x + 36
Isolate Constants Now, subtract 3 x from both sides: 36 = 36
Determine the number of solutions Since the variables have been eliminated and we are left with a true statement, this equation has infinitely many solutions.
Examples
Imagine you're trying to balance a seesaw. If no matter where you and your friend sit, the seesaw remains perfectly balanced, it means there are infinitely many solutions to the balancing act. Similarly, in our equation, no matter what value we choose for x , the equation will always hold true, indicating infinitely many solutions.
