Problem 61

Question

Simplify each algebraic expression. $$12+5(3 x-2)$$

Step-by-Step Solution

Verified

Answer

The simplified form of the algebraic expression $12+5(3x-2)$ is $15x + 2$.

1Step 1: Distribute Multiplication

In the expression $12+5(3x-2)$, first distribute the $5$ across the $3x-2$, by multiplying $5$ with $3x$ and then $5$ with $-2$. This will give: $12+15x-10$.

2Step 2: Addition

Now, perform the addition operation by adding the constant terms $12$ and $-10$. This will give a simplified expression: $15x + 2$.

Key Concepts

Distribution in AlgebraCombining Like TermsAlgebraic Expressions

Distribution in Algebra

In algebra, distribution refers to the Distributive Property, which lets you multiply a single term by terms inside a parenthesis. This is a crucial step for simplifying algebraic expressions. When you see something like $ a(b + c) $, it means you have to multiply $ a $ by each term inside the parentheses: $ a \times b $ and $ a \times c $. This results in the expression: $ ab + ac $. This property is incredibly helpful, particularly when dealing with polynomials or expressions with variables.
This method breaks down complex multiplication into smaller and easier steps. For instance, in the expression $12+5(3x-2)$, we apply distribution by multiplying $5$ with both $3x$ and $-2$.
Thus, it simplifies to $12 + 15x - 10$, making other operations straightforward.

Combining Like Terms

Once you have distributed the terms in an algebraic expression, the next step is to simplify further by combining like terms. Like terms are terms that contain the same variables raised to the same power. They can be constants, like numbers, or terms with variables.
For instance, in the expression $15x + 12 - 10$, we identify the like terms as the constants $12$ and $-10$. These are the terms we can "combine," meaning to add or subtract them to reduce the expression to its simplest form.

Combine constants or numbers: $12 - 10 = 2$.
Keep the variable terms as they are unless they have like terms to combine with.

Thus, by combining $12$ and $-10$, we simplify the expression to $15x + 2$.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are used to express mathematical relationships and solve problems. An expression can be as simple as a single number or as complex as a polynomial with multiple variables and operations.
In learning how to manipulate these expressions, it is crucial to understand components such as terms, coefficients, and constants. A term is a part of an expression that can include numbers, variables, or both.

For example, in $15x + 2$, $15x$ is a term and $2$ is another term.
A coefficient is the numerical part of a term that has a variable, like the "15" in $15x$.
A constant is a term without a variable, like the "2" here.

Understanding these components makes it easier to simplify and solve expressions, as it helps to identify which operations to perform, such as distributing and combining like terms.

Problem 61

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