Answered

Kate begins solving the equation $\frac{2}{3}(6 x-3)=\frac{1}{2}(6 x-4)$. Her work is correct and is shown below.



$\begin{aligned}

\frac{2}{3}(6 x-3) & =\frac{1}{2}(6 x-4) \\

4 x-2 & =3 x-2

\end{aligned}$



When she adds 2 to both sides, the equation $4 x=3 x$ results. Which statement best describes what happens to $x$?


AThe equation has infinite solutions.
BThe equation has one solution: $x=0$.
CThe equation has one solution: $x=\frac{4}{3}$.
DThe equation has no solution.

Verified Answer :

  • The equation simplifies from \(\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)\) to \(4x - 2 = 3x - 2\).
  • After adding 2 to both sides, the equation becomes \(4x = 3x\).
  • Subtracting \(3x\) from both sides isolates the variable to find \(x = 0\).
  • The equation has one solution: \(x = 0\), which is \(\boxed{\text{The equation has one solution: } x = 0}\).

Explanation

  1. Analysis of the Problem

We are given the equation \(\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)\) and Kate's simplified version \(4x - 2 = 3x - 2\). When Kate adds 2 to both sides, she arrives at the equation \(4x = 3x\). We need to determine the value of \(x\) that satisfies this equation and identify the correct description of its solutions from the given choices.

  1. Solving and Verifying the Equation

Starting with the equation \(4x = 3x\), we can isolate the variable \(x\) by moving all terms containing \(x\) to one side. Subtract \(3x\) from both sides of the equation:
\[4x - 3x = 0\]
\[x = 0\]
This shows that \(x = 0\) is a potential solution. To verify this, substitute \(x = 0\) back into the original equation:
Left-hand side (LHS): \(\frac{2}{3}(6(0) - 3) = \frac{2}{3}(-3) = -2\)
Right-hand side (RHS): \(\frac{1}{2}(6(0) - 4) = \frac{1}{2}(-4) = -2\)
Since the LHS equals the RHS, \(x = 0\) is a valid solution. Because the equation \(x = 0\) is a linear equation with a single variable, it has exactly one solution.

  1. Conclusion

We have determined that the only value of \(x\) that satisfies the equation is \(x = 0\). Comparing this to the given options:

  • The equation has infinite solutions: Incorrect.
  • The equation has one solution: \(x = 0\): Correct.
  • The equation has one solution: \(x = 4/3\): Incorrect.
  • The equation has no solution: Incorrect.

Thus, the equation has one solution, \(x = 0\).

Examples

Understanding how to solve simple linear equations is crucial for balancing budgets or determining the cost of items in a shopping list. For example, if you are comparing two different pricing models for a service, you might set up an equation where the two models are equal. Finding the solution to such an equation, much like solving for \(x\) in our problem, helps you decide which option is more cost-effective or if they are equivalent at a specific level of usage.