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 solution will best illustrate 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 :

  • Simplify the given equation \(\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)\) to \(4x - 2 = 3x - 2\).
  • Add 2 to both sides to obtain the equation \(4x = 3x\).
  • Subtract \(3x\) from both sides to isolate the variable, resulting in \(x = 0\).
  • Conclude that the equation has one solution: \(x = 0\).
  • Final Answer: \(\boxed{x = 0}\)

Explanation

  1. Analysis of the Equation

We are analyzing Kate's equation solving process:
\(\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)\).
By distributing the fractions on both sides, she correctly simplified this to:
\(4x - 2 = 3x - 2\).
Adding \(2\) to both sides leads to the simplified equation:
\(4x = 3x\).
Our objective is to determine the solution set for \(x\) based on this final linear equation.

  1. Solving the Equation for x

To solve the equation \(4x = 3x\), we need to isolate the variable \(x\). We can achieve this by subtracting \(3x\) from both sides of the equation:
\(4x - 3x = 3x - 3x\)
\(x = 0\).
This shows that there is exactly one solution to the equation, which is \(x = 0\). We can verify this by substituting \(x = 0\) back into the original equation:
Left-hand side: \(\frac{2}{3}(6(0) - 3) = \frac{2}{3}(-3) = -2\)
Right-hand side: \(\frac{1}{2}(6(0) - 4) = \frac{1}{2}(-4) = -2\)
Since both sides are equal, the solution \(x = 0\) is correct.

  1. Conclusion

Based on the derivation, the equation simplifies to \(x = 0\), which indicates there is exactly one solution. Comparing this to the given options, the correct statement is that the equation has one solution: \(x = 0\).

Examples

Solving linear equations is a fundamental skill used in everyday life for tasks such as calculating budgets, determining the time needed for travel based on distance and speed, or adjusting recipes based on the number of people serving. For instance, if you have a budget of \$20 and you are buying ingredients for a dish that costs a base fee plus a per-item price, you set up an equation to find how many items you can afford. The logic used here to isolate the variable is exactly the same as finding that unknown quantity in any real-world problem.

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