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In algebra, coefficients determine how much of a variable is present in an equation. The concept of negative coefficients often puzzles beginners. A negative coefficient indicates that the variable it multiplies is subtracted from something or has a negative direction.

Consider the equation \(-x = -1 \cdot x\). Here, \(-x\) is another way of saying \(-1 \cdot x\). This means we are multiplying the variable \(x\) by a coefficient of \-1.

When you encounter a negative coefficient, remember it simply flips the direction or sign of the variable it is associated with.

• If you see \(-3x\), it equates to subtracting \,3\cdot x\ from zero or a start point.
• This is as opposed to \(3x\), which would imply adding \,3\cdot x\.

Understanding this helps make sense of why in the equation from the exercise, the blank was filled with \-1.

Dividing by integers is a common task in algebra, often necessary to simplify expressions or solve equations. When dividing a term by an integer, you're essentially scaling down the value of that term by that integer factor.

Take the example \(\frac{2t}{3}\). Here, we divide the product of \(t\) by 3. It translates to \(\frac{2}{3} \cdot t\), simplifying the expression while altering the magnitude of \(t\).

This operation highlights that dividing an entire expression by a number is the same as multiplying each constituent by the reciprocal of that number.

• In simpler terms, if you have \(\frac{t}{a}\), this could be rewritten as \(\frac{1}{a} \cdot t\).
• Remember to apply this concept each time you encounter fractions in an equation.

Therefore, the blank was correctly filled with \(\frac{2}{3}\) in the original exercise.

One of the most crucial steps in solving algebra equations is verifying the solution you obtain. Checking solutions is not just a formality but a necessary step to ensure your equation is solved accurately.

To check a solution, substitute the found value back into the original equation to see if both sides of the equation remain equal.

Let’s consider part (a) of the example. After understanding that \(-x\) is equal to \(-1 \cdot x\), substitute within the original equation. If both sides match, your solution is validated. Similarly, for part (b), replace the variable in the form \(\frac{2t}{3} = \frac{2}{3} \cdot t\). Does the equality hold? If yes, your solution is correct.

• This method prevents mistakes and ensures that no step in your calculation process was mishandled.
• Develop the habit of checking your results as it will ensure that solutions remain consistent and correct.

By integrating this practice, you boost your problem-solving confidence.