Problem 82 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(3(2 y-5)-4(5 y-7)\)

Step-by-Step Solution

Answer

-14y + 13

Step 1 - Distribute the Constants

Distribute the constants 3 and -4 into the expressions inside the parentheses. This gives: \(3(2y) - 3(5) - 4(5y) - 4(-7)\)

Step 2 - Simplify Each Term

Multiply the constants with each term inside the parentheses: \(6y - 15 - 20y + 28\)

Step 3 - Combine Like Terms

Combine the like terms (terms with the same variable): \(6y - 20y - 15 + 28\). This simplifies to: \(-14y + 13\)

Key Concepts

distributive property

The distributive property is a core concept in algebra that makes solving expressions easier. This property allows you to multiply a single term by each term inside a set of parentheses. When we say distribute, we mean to apply multiplication over addition or subtraction inside the parentheses.

For instance, in the expression given: 3(2y - 5), we apply the distributive property as follows:

• Multiply 3 by 2y to get 6y.
• Multiply 3 by -5 to get -15.

So, 3(2y - 5) becomes 6y - 15.

Similarly, for -4(5y - 7):

• Multiply -4 by 5y to get -20y.
• Multiply -4 by -7 to get 28.

Thus, -4(5y - 7) becomes -20y + 28. This shows how the distributive property helps in breaking down complex expressions into simpler parts that we can work with more easily.

combining like terms

Combining like terms involves adding or subtracting terms that have the same variable parts in an algebraic expression. This step helps to simplify expressions further.

Let’s look at our example where we have obtained the terms: 6y - 15 - 20y + 28. We can combine like terms by grouping terms with the same variable.

Here are the steps:

• Combine the 'y' terms: 6y - 20y.
• Combine the constant numbers: -15 + 28.

So, 6y - 20y simplifies to -14y, and -15 + 28 simplifies to 13.

This results in the simplified expression: -14y + 13.

It's important to note that only the coefficients (numerical parts) of like terms are added or subtracted while the variable part remains unchanged.

simplification of algebraic expressions

Simplifying algebraic expressions involves applying a series of steps to condense expressions to their simplest form. This usually requires using the distributive property and combining like terms.

To simplify the original expression 3(2y - 5) - 4(5y - 7), we:

• First use the distributive property to expand the expression: 3 * 2y - 3 * 5 - 4 * 5y - 4 * -7.
• Simplify each term: 6y - 15 - 20y + 28.
• Combine the like terms: 6y - 20y, which simplifies to -14y; and -15 + 28, which simplifies to 13.

Finally, our expression is fully simplified to: -14y + 13.

Simplification is essential in making expressions easier to work with, especially for solving equations or further algebraic manipulation. It enables us to see relationships between terms more clearly and prepares us for more advanced mathematical operations.