Problem 27 - Elementary and Intermediate Algebra

Question

To prepare for Section 9.3, review simplifying expressions \((\text { Section } 1.8)\) Simplify. [ 1.8] $$ -2(2 x-3 y) $$

Step-by-Step Solution

Answer

-4x + 6y

Step 1: Distribute the -2

Apply the distributive property: multiply each term inside the parentheses by -2.

Step 2: Multiply -2 by 2x

Perform the multiplication: -2(2x) = -4x.

Step 3: Multiply -2 by -3y

Perform the multiplication: -2(-3y) = 6y.

Step 4: Combine the results

Combine the results from the previous multiplications to get the simplified expression: -4x + 6y.

Key Concepts

Distributive Property

The distributive property helps us multiply a single term by each term inside a set of parentheses. In this exercise, we have the expression -2(2x - 3y). By using the distributive property, we need to multiply -2 by both 2x and -3y.This means we distribute the -2 to both terms, changing the original expression to: -2 * 2x and -2 * -3y. Let's explore what happens next.

Multiplication of Terms

Now, we use multiplication to simplify our expression further. As we apply the distributive property, we get two separate multiplications: -2 * 2x and -2 * -3y.When we multiply -2 by 2x, we get: -2 * 2x = -4x Next, we multiply -2 by -3y: -2 * -3y = 6y.Remember, multiplying two negative numbers results in a positive number, which is why we get positive 6y. By breaking the expression into smaller steps, we can manage the multiplications easily.

Combining Like Terms

After distributing and multiplying, the expression becomes: -4x + 6y. In this case, there are only two terms: -4x and 6y. Each term is different, so we cannot combine them further. Combining like terms means adding or subtracting terms that have the same variable and exponent. Here, we do not have any like terms, so our final simplified expression remains -4x + 6y.

Negative Coefficients

Handling negative coefficients correctly is key to simplifying expressions. In our initial exercise, we began with -2(2x - 3y). The -2 is a negative coefficient that we used in the distributive property. A negative coefficient changes the sign of each term it multiplies. In our example: -2 * 2x = -4x and -2 * -3y = 6y.The negative coefficient -2 changed the sign from - to + when multiplying -3y. By properly handling negative coefficients, we simplify expressions accurately every time.