Problem 61 - Elementary Algebra

Question

\(3(2 x+1)-7(4 x-9)\)

Step-by-Step Solution

Answer

-22x + 66

Step 1: Distribute the 3

Multiply 3 by each term inside the first set of parentheses: 3(2x + 1) = 6x + 3

Step 2: Distribute the -7

Multiply -7 by each term inside the second set of parentheses: -7(4x - 9) = -28x + 63

Step 3: Combine the results

Add the results from the previous steps together: 6x + 3 - 28x + 63

Step 4: Combine like terms

Combine the x terms and the constants separately: (6x - 28x) + (3 + 63) = -22x + 66

Key Concepts

Distributive Property

The distributive property is a key concept in algebra. It allows you to simplify expressions by multiplying each term inside parentheses by a factor outside the parentheses. In our example, we start with the expression \[3(2x+1)-7(4x-9)\]Using the distributive property, we distribute 3 to each term inside the first set of parentheses:\[3(2x + 1) = 6x + 3\]Next, we distribute -7 to each term inside the second set of parentheses:\[-7(4x - 9) = -28x + 63\]These steps help break down the expression into simpler parts.

Combining Like Terms

Combining like terms is an essential step when simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. In our example, after using the distributive property, we have:\[6x + 3 - 28x + 63\]Here, you can see the terms involving 'x' (6x and -28x) and the constant terms (3 and 63). To combine like terms, group and add or subtract them:\[(6x - 28x) + (3 + 63)\]This helps us in further simplifying the expression.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. This involves performing arithmetic operations and combining like terms. After applying the distributive property and identifying like terms in our example, we have:\[6x + 3 - 28x + 63\]Next, combine like terms as we did earlier:\[(6x - 28x) + (3 + 63)\]Carry out the arithmetic:\[-22x + 66\]This is the simplest form of the given expression.

Elementary Algebra

Elementary algebra lays the foundation for understanding and manipulating algebraic expressions. Concepts such as the distributive property, combining like terms, and simplifying expressions are fundamental. In our example, we started with\[3(2x+1)-7(4x-9)\]We used the distributive property to expand the expression:\[6x + 3 - 28x + 63\]Then we combined like terms and simplified to get:\[-22x + 66\]Mastering these basic concepts is crucial for solving more complex algebraic problems in the future.