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Chapter 2: Problem 6

Multiply. $$ -35(10 x-5) $$

Short Answer

Expert verified
The expression simplifies to \(-350x + 175\).

Step by step solution

01

Distribute the Negative Sign

We begin by distributing the negative sign attached to -35 into the expression inside the parentheses. This can be rewritten as \[ -35 \times (10x - 5) = -35 \times 10x + (-35) \times (-5) \].
02

Multiply the First Term

Multiply -35 by 10x to get the first product. \[ -35 \times 10x = -350x \].
03

Multiply the Second Term

Now multiply -35 by -5 to get the second product. Remember that multiplying two negative numbers results in a positive product. \[ -35 \times (-5) = 175 \].
04

Combine Results

Combine the products from Step 2 and Step 3 to get the final expression. \[ -350x + 175 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in algebra that helps simplify expressions. It states that multiplying a single term by a whole expression inside parentheses involves distributing the multiplication to each term within. This means when you have an expression like \(-35(10x - 5)\), you need to multiply \-35\ with every term inside the parentheses.

Here's a simple way to visualize it:
  • Think of distributing as sharing, where you share the multiplier with each part of the expression inside the parentheses.
  • In \(-35(10x - 5)\), share \-35\ with \10x\ and \5\ individually.
This results in \(-35 imes 10x\) and \(-35 imes -5\).

Remember, the distributive property helps break down complex expressions into manageable parts, making it easier to perform calculations.
Negative Numbers
Negative numbers can be tricky, but they follow specific rules in multiplication. When you multiply a negative number by another number, the results can vary:

  • Negative times positive gives a negative result: \(-35 imes 10x = -350x\).
  • Negative times negative results in a positive: \(-35 imes -5 = 175\).
Understanding these rules is crucial for solving algebraic problems accurately.

Think of negative numbers as the opposite or reverse. Multiplying two negatives, think of reversing twice, which takes you back to a positive position. Knowing these rules helps ensure the correctness of your solutions.
Multiplying Terms
Multiplying terms involves combining like terms through multiplication. This process can include variables, like \x\, and coefficients, like \-35\.

Here's how it works:
  • Apply multiplication to coefficients: \(-35 imes 10 = -350\).
  • Carry along the variable when multiplying: \(-350 imes x = -350x\).
When multiplying terms, keep track of the coefficients and the variables. Always multiply coefficients together first, then apply the result to the variables.

Combining these techniques helps simplify expressions and solve equations, making it a vital skill in algebra.

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