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When solving proportions like \( \frac{2}{3}=\frac{x}{6} \), cross-multiplication is a powerful method to use. Proportions represent an equation stating two ratios are equal. Cross-multiplication helps eliminate the fractions, making it easier to solve for the unknown variable.
To apply cross-multiplication, you multiply the numerator of the first fraction by the denominator of the second fraction. Then, do the reverse: multiply the numerator of the second fraction by the denominator of the first fraction. Here it becomes:
\[ 2 \times 6 = 3 \times x \]
Cross-multiplying gives us:\[ 12 = 3x \]
Now, you're left with a simple equation that no longer involves fractions, making it straightforward to solve for \( x \).

After cross-multiplying, the next step in solving proportions is to isolate the variable. This means getting \( x \) by itself on one side of the equation. For the equation \( 12 = 3x \), you'll divide both sides by 3 to solve for \( x \). This division step is crucial:
\[ \frac{12}{3} = \frac{3x}{3} \]
Simplifying the equation yields:
\[ x = 4 \]
Isolating the variable allows you to find the specific value that satisfies the original proportion. Always make sure that you perform the same operation on both sides of the equation to maintain the balance of the equation.

Verification is an essential step after solving the equation. It ensures that the solution found is correct. Substitute the value of \( x \) back into the original proportion to check it holds true. In our example, substitute \( x = 4 \) back into \( \frac{2}{3}=\frac{x}{6} \):
\[ \frac{2}{3} = \frac{4}{6} \]
Now, simplify \( \frac{4}{6} \) and you'll find that it equals \( \frac{2}{3} \), confirming the solution is correct.

• Checking your work helps you verify the steps taken were correct.
• It ensures that no calculation errors were made.

Verification strengthens your reliability in solving mathematical problems and increases your confidence in the results.