Problem 58 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ (3 x-2+6 y)+(7 x-2-y) $$

Step-by-Step Solution

Answer

The sum is \(10x - 4 + 5y\).

Step 1: Distribute Parentheses

Begin by rewriting the expression without the parentheses. Since addition is involved, you can directly remove the parentheses: \((3x - 2 + 6y) + (7x - 2 - y) = 3x - 2 + 6y + 7x - 2 - y\).

Step 2: Combine Like Terms (x terms)

Identify like terms involving \(x\). Combine \(3x\) and \(7x\): \(3x + 7x = 10x\).

Step 3: Combine Like Terms (constant terms)

Identify constant terms. Combine \(-2\) and \(-2\): \(-2 - 2 = -4\).

Step 4: Combine Like Terms (y terms)

Identify like terms involving \(y\). Combine \(6y\) and \(-y\): \(6y - y = 5y\).

Step 5: Write Final Expression

Now, put all the combined terms together to create the final expression: \(10x - 4 + 5y\).

Key Concepts

Like Terms

Understanding "like terms" is essential when dealing with polynomials and algebraic expressions. Like terms are terms with the same variable raised to the same power. For example, in the expression \(3x - 2 + 6y + 7x - 2 - y\), the "like terms" for \(x\) are \(3x\) and \(7x\). Similarly, for the variable \(y\), the like terms are \(6y\) and \(-y\), whereas the constants \(-2\) can also be considered like terms. When you want to combine like terms, you simply add or subtract their coefficients and keep the variable part unchanged. It's like gathering all your pennies if you're counting change — you simply collect them together! This concept helps simplify expressions and is a foundational skill in algebra.

Distributive Property

The distributive property is an important algebraic principle used to simplify expressions that include parentheses. This property tells us how to deal with expressions involving multiplication over addition or subtraction. For example, the expression \(a(b + c)\) can be expanded as \(ab + ac\). In our exercise, we had the expression \((3x - 2 + 6y) + (7x - 2 - y)\). With addition, there’s no need to distribute any multiplication; however, it’s crucial to remove the parentheses effectively. Here, since adding doesn’t affect the terms inside, we omit the parentheses: \(3x - 2 + 6y + 7x - 2 - y\). Understanding how to apply the distributive property correctly ensures accurate simplification of expressions, which leads to the correct combination of like terms.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators (like \(+\) and \(-\)). They are the building blocks of algebra and allow us to generalize mathematical relationships. These expressions can be simple, like \(3x\), or more complex, such as \(3x - 2 + 6y\).In the given exercise, we work with such expressions. Each term in an algebraic expression usually consists of a coefficient (a number), a variable (like \(x\) or \(y\)), and an exponent if applicable. By understanding algebraic expressions, we learn to manipulate and rewrite them, which is essential for solving equations and understanding more advanced math topics. Practicing with various algebraic expressions helps strengthen our ability to identify and work with each component.

Combining Terms

Combining terms in an algebraic expression involves adding or subtracting like terms to simplify the expression. By grouping terms with the same variable part, we make expressions easier to understand and solve. For example, in the expression \(3x - 2 + 6y + 7x - 2 - y\), we need to find and merge like terms:

• Combine \(x\) terms: \(3x + 7x = 10x\)
• Combine constant terms: \(-2 - 2 = -4\)
• Combine \(y\) terms: \(6y - y = 5y\)

After combining like terms, the expression becomes \(10x - 4 + 5y\). This process of combining like terms reduces the complexity of the expression and is a vital skill for solving equations effectively. Regular practice ensures better fluency in simplifying expressions in algebra.