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To solve rational equations like \( \frac{q-2}{2}=\frac{2q-7}{18} \), we start by using cross-multiplication. This technique helps eliminate the denominators (numbers below the fraction line).When we cross-multiply, we multiply the numerator of one fraction by the denominator of the other fraction. For our exercise, this means multiplying \( q-2 \) by 18 and \( 2q-7 \) by 2.So, it looks like this:\[ (q-2) \times 18 = (2q-7) \times 2 \]We have essentially turned our initial problem into a simpler equation without fractions, making it easier to solve.

After cross-multiplying, our equation is:\[ 18(q-2) = 2(2q-7) \]Next, we need to isolate the variable 'q'. This means getting 'q' all by itself on one side of the equation. First, let’s expand the equation by performing the multiplications:\[ 18q - 36 = 4q - 14 \]To isolate 'q', we need all the 'q' terms on one side and the constant numbers on the other side. We move all 'q' terms from one side of the equation to the other by subtracting or adding. In our case, we subtract 4q from both sides to get:\[ 18q - 4q = -14 + 36 \]This simplifies to:\[ 14q = 22 \]We are almost done. To completely isolate 'q', we simply divide both sides by 14:\[ q = \frac{22}{14} \]Which gives:\[ q = \frac{11}{7} \]Our variable 'q' is now alone, and we know its value.

Simplifying equations is about making them easier to solve.First, we dealt with cross-multiplication to remove fractions. This transformed our original equation into:\[ 18(q-2) = 2(2q-7) \]Then, we expanded and simplified it to:\[ 18q - 36 = 4q - 14 \]By moving and combining terms, we further simplified it to:\[ 14q = 22 \]Finally, we simplified it one last time by dividing both sides to isolate 'q':\[ q = \frac{11}{7} \]Remember:- Always perform the same operations on both sides of the equation to keep it balanced.- Combine like terms where possible.- Simplify in multiple steps if needed; don’t rush.This careful process of rearranging and combining terms ensures that our final value for 'q' is correct and in its simplest form.