91Ó°ÊÓ

The distributive property is a fundamental algebraic principle used to simplify expressions. It states that for any numbers or variables a, b, and c, the equation: \[ a(b + c) = ab + ac \] applies. In simpler terms, you distribute the term outside the parentheses to each term inside the parentheses.

Let's look at the example given in the exercise: \[ -1(2x - 7) \] Step 1 involves using the distributive property. We distribute \[ -1 \] to each term inside the parentheses: Hence \[ -1 \times (2x - 7) \] which becomes \[ -1 \times 2x - 1 \times -7 \].

By applying the distributive property correctly, we ensure that every term inside the parentheses is multiplied by the term outside, simplifying our original equation.

Simplification in algebra involves combining like terms and performing basic arithmetic operations to make expressions easier to understand and handle. In the next step of our example, we perform the individual multiplications.

Our distributed terms are: \[ -1 \times 2x \] and \[ -1 \times -7 \]. For the first multiplication:\[ -1 \times 2x = -2x \]For the second multiplication:\[ -1 \times -7 = 7 \].

After calculating the products, we combine the simplified terms. In our case: \[ -2x + 7 \]. Now, the expression is fully simplified.

Remember, breaking down the process into clear steps like these will help in efficiently solving even more complex algebraic expressions.

Understanding how to handle negative numbers is crucial in algebra. Negative numbers require careful attention, especially during multiplication and addition/subtraction.

Let's go back to our example to see this in action: \[ -1 \times (2x - 7) \]. When we multiply \[ -1 \] by \[ 2x \], we get: \[ -2x \]. Multiplying a negative number by a positive number always results in a negative outcome.

Next, consider the multiplication \[ -1 \times -7 \]. When we multiply two negative numbers, the result is positive, thus \[ -1 \times -7 = 7 \].

By understanding and correctly handling the rules for negative numbers, we can simplify expressions without errors. Always remember to check the signs to ensure accuracy!