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The distributive property is a useful tool in algebra. It lets you multiply a single term by each term inside parentheses.
In this exercise, you come across an equation that uses the distributive property: \(0.3(30) + 0.15x = 0.2(30 + x)\).
You need to distribute the constants (0.3 and 0.2) to each term inside the parentheses.
This gives you: \[0.3 \times 30 + 0.15x = 0.2 \times 30 + 0.2x\].
After carrying out the multiplication, you get: \[9 + 0.15x = 6 + 0.2x\].
This step very clearly shows how the distributive property helps in simplifying the given equation by removing parentheses and spreading the constants across the terms.

Isolating the variable is crucial to solving any algebraic equation.
It means getting the variable (in this case, x) alone on one side of the equation.
After distributing, you have the equation: \[9 + 0.15x = 6 + 0.2x\].
To isolate x, begin by moving all terms with x to one side. Subtract 0.15x from both sides: \[9 + 0.15x - 0.15x = 6 + 0.2x - 0.15x\].
This simplifies to: \[9 = 6 + 0.05x\].
Next, move the constant term on the side with the variable by subtracting 6 from both sides: \[9 - 6 = 6 - 6 + 0.05x\].
This simplifies further to: \[3 = 0.05x\].
To get x by itself, divide both sides by 0.05: \[x = \frac{3}{0.05}\].
Doing the division, you find: \[x = 60\].
Isolating the variable makes the equation easy to solve.

Checking your solutions ensures that you haven't made any mistakes along the way.
After finding \(x = 60\), substitute it back into the original equation: \[0.3(30) + 0.15(60) = 0.2(30 + 60)\].
Simplify both sides:
Left-hand side: \[0.3 \times 30 + 0.15 \times 60 = 9 + 9 = 18\].
Right-hand side: \[0.2 \times 90 = 18\].
Since both sides are equal: \[18 = 18\], your solution is correct.
Always rechecking helps solidify your understanding and ensures accuracy in problem-solving.