Problem 33 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ -(3 x-1)-2(5 x-1)+4(-2 x-3) $$

Step-by-Step Solution

Answer

The simplified expression is \(-21x - 9\).

Step 1: Distribute Negative Sign

Apply the negative sign in front of the first parentheses to each term inside: \(-1(3x - 1) = -3x + 1\). Replace the original expression: \(-(3x - 1) = -3x + 1\).

Step 2: Distribute -2 into the Second Parenthesis

Multiply \(-2\) by each term inside the second parentheses: \(-2(5x - 1) = -10x + 2\). Replace in the expression to get: \(-3x + 1 - 10x + 2\).

Step 3: Distribute 4 into the Third Parenthesis

Multiply \(4\) by each term inside the third parentheses: \(4(-2x - 3) = -8x - 12\). Add this to the expression so far: \(-3x + 1 - 10x + 2 - 8x - 12\).

Step 4: Combine Similar Terms

Combine all the terms with \(x\): \(-3x - 10x - 8x = -21x\). Next, combine all the constant terms: \(1 + 2 - 12 = -9\).

Step 5: Write the Final Expression

The simplified expression is obtained by combining the results from previous steps: \(-21x - 9\).

Key Concepts

Distributing Terms

When simplifying algebraic expressions, distributing terms is a crucial skill to master. This technique involves spreading a multiplication over terms enclosed in parentheses. For example, the expression \(3(a + b)\) means that the multiplier 3 is applied to both \(a\) and \(b\). Thus, \(3(a + b) = 3a + 3b\).
This method ensures that each term inside the parentheses is correctly multiplied by the factor outside, maintaining proper order of operations. Distributing terms carefully sets the stage for combining like terms in the next steps.

Combining Like Terms

After distributing terms, the next step in simplifying algebraic expressions is to combine like terms. Like terms are terms that have the same variable and exponent. For example, \(2x\) and \(5x\) are like terms because they both involve the variable \(x\). However, \(2x\) and \(3y\) are not like terms.
Combining like terms involves adding or subtracting them, which helps simplify the expression further. In the example \(-3x - 10x - 8x\), these are all like terms with the same variable, \(x\). By combining them, you simplify the expression to \(-21x\).

• Ensure all similar terms are combined into a single term.
• This step reduces clutter in the expression, making it simpler to understand and use.

Algebraic Expressions

Algebraic expressions are a fundamental component in algebra, consisting of constants, variables, and operations. An algebraic expression like \(7x + 3 - 2y\) consists of terms—the parts separated by addition or subtraction. Each term is either a number, a variable, or a product of numbers and variables.
Understanding algebraic expressions involves:

• Recognizing parts of the expression, such as coefficients, variables, and constants.
• Applying operations like addition, subtraction, distribution, and combining terms.

Mastering the manipulation of these expressions is essential for solving equations and simplifying complex mathematical problems.

Negative Signs in Algebra

Handling negative signs in algebra correctly is vital to simplifying expressions properly. Negative signs can affect entire terms, and misunderstanding them can lead to incorrect results.
In the expression \(-(3x - 1)\), the negative sign applies to the whole term inside the parentheses. Distributing it yields \(-3x + 1\), which is a critical step in simplification.
Here are some tips for working with negative signs:

• Always distribute negative signs correctly across all terms in parentheses.
• Remember that subtracting a negative is the same as adding a positive.

Being careful with negative signs ensures the expression retains its correct value.