Problem 81 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(2(5 r+3)-3(2 r-3)\)

Step-by-Step Solution

Answer

4r + 15

Step 1 - Distribute

First, distribute the constants outside the parentheses: Distribute the 2 in the first term: 2(5r + 3) This becomes: 10r + 6Now, distribute the -3 in the second term:-3(2r - 3)This becomes:-6r + 9

Step 2 - Combine Like Terms

Combine the like terms from the distributed terms. You have two terms with the variable r (10r and -6r) and two constant terms (6 and 9): 10r - 6r + 6 + 9 Combine the terms with r: 4rCombine the constant terms:15

Step 3 - Write Final Expression

Combine the simplified terms to form the final simplified expression. You have already combined like terms to get 4r and 15, so combine these to get:4r + 15

Key Concepts

Distribution

The first step in simplifying an algebraic expression is often distribution. Distribution means multiplying each term inside the parentheses by the factor outside the parentheses. For example, in the expression \(2(5r + 3)\), we distribute the 2 to both 5r and 3.
This gives us: \(2 \times 5r + 2 \times 3\)
Simplified, that becomes: \(10r + 6\)
Next, look at the second part of our original expression: \(-3(2r - 3)\). Distribute the -3 to both 2r and -3. This gives us: \(-3 \times 2r + (-3) \times (-3)\).
Simplified, this becomes: \(-6r + 9\).
Understanding distribution helps break down complex expressions into simpler parts, making it easier to handle the remaining steps.

Like Terms

After distributing, you will encounter 'like terms.' Like terms are terms that contain the same variable raised to the same power. You can combine like terms to simplify the expression.
In our example, after distribution, we have: \(10r + 6 - 6r + 9\).

• Terms with the variable r: \(10r and -6r\)
• Constant terms: \(6 and 9\)

To combine like terms:
1. Combine the r-terms: \(10r - 6r = 4r\)
2. Combine the constant terms: \(6 + 9 = 15\).
By recognizing and combining like terms, we make the expression simpler and more manageable.

Simplifying Expressions

Simplifying an expression means reducing it to its most basic form. Simplification helps in better understanding and solving algebraic problems. Following our example, after distributing and combining like terms, we have: \(4r + 15\).
This is the simplest form of the given expression.

• It has no parentheses.
• It combines all like terms.

Simplifying expressions is like cleaning up a messy room. It makes it easier to see what's there and to work with it. Always remember the steps:
1. Distribute.
2. Combine like terms.
3. Write the simplified expression.
Mastering these steps will make you proficient in handling various algebraic problems.