Verified Answer :

  • Simplify the left-hand side to \(-3x + 5\).
  • Set the simplified left-hand side equal to the right-hand side: \(-3x + 5 = 3x + 5\).
  • Subtract 5 from both sides and combine like terms to get \(6x = 0\).
  • Solve for \(x\) to find the unique solution \(x = 0\), meaning there is \(\boxed{\text{One Solution}}\).

Explanation

  1. Simplify the Equation

We start by simplifying the given equation:
\[2x + 7 - 5x - 2 = 3x + 5\]
First, we combine the like terms on the left-hand side. The terms containing \(x\) are \(2x\) and \(-5x\), which sum to \(-3x\). The constant terms are \(7\) and \(-2\), which sum to \(5\). Thus, the equation simplifies to:
\[-3x + 5 = 3x + 5\]

  1. Isolate the Variable \(x\)

Next, we aim to isolate the variable \(x\). We can subtract \(5\) from both sides of the equation:
\[-3x = 3x\]
Now, we add \(3x\) to both sides to gather all terms involving \(x\) on one side:
\[0 = 3x + 3x\]
\[0 = 6x\]

  1. Determine the Number of Solutions

Finally, we solve for \(x\) by dividing both sides by \(6\):
\[x = \frac{0}{6}\]
\[x = 0\]
Since we found a single, unique value for \(x\), the equation has exactly one solution.

Examples

Suppose you are trying to determine if two different companies have a break-even point at the same production volume. You set up two linear cost functions and equate them. Solving this equation for the volume variable tells you if there is a single point where costs align (one solution), if they are never equal (no solution), or if they are identical (infinite solutions). Finding the exact point where they intersect helps in decision-making and optimization.

Related Questions