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Working with fractions in equations can make the problem look more complicated than it actually is. To make the process easier, it helps to clear the fractions right away. This involves getting rid of the fraction by multiplying every term in the equation by the denominator. In our example, we had \[\frac{1}{3}(7 - 7x) = -21\]To clear the fraction, multiply both sides of the equation by 3 (the denominator). So you get the equation:\[3 \times \frac{1}{3} (7 - 7x) = 3 \times (-21)\] This step ensures the fraction is cleared, leading to a simpler equation:\[7 - 7x = -63\]When every term is multiplied appropriately, the fractions disappear and you are left with a cleaner equation. Always double-check each multiplication to avoid mistakes. This step helps to simplify the equation at a very early stage, setting you up for easier calculations in the following steps.

Once you've cleared any fractions, it's important to isolate the terms you want to solve for. In most linear equations, this means getting the variable (like \(x\)) by itself on one side of the equation. In our example:\[7 - 7x = -63\]You'll want to focus on the term containing \(x\). Start by subtracting the constant term (in this case, 7) from both sides. This step is crucial because it separates the variable part from the constant values. So, subtract 7 from both sides:\[7 - 7x - 7 = -63 - 7\]This simplifies to:\[-7x = -70\]The goal here is to create a clear path to finding \(x\). By isolating the \(x\)-term, you're simplifying the situation, making it easier to see what needs to be done next. Remember, every operation should be done to both sides of the equation to maintain balance.

After isolating the variable term, you often need one last action to solve for it. This is particularly true if the variable term has a coefficient. Simplifying means making the equation cleaner and easier to solve: in essence, reducing it to its simplest form. Currently, we have:\[-7x = -70\]To find \(x\), divide both sides by the coefficient of \(x\) (which is -7). Division helps to cancel out the -7 next to \(x\), isolating the variable:\[x = \frac{-70}{-7}\]This simplifies to:\[x = 10\]Simplification ensures you've reached the solution in its neatest form. It's about breaking down the steps so every move is logical and clear. The final value of \(x\) should make sense when you consider the entire equation, ensuring there's consistency in the arithmetic used throughout the solution.