Problem 49 - Elementary Algebra

Question

Solve. $$ -6(x-1)-3 x-3(x+8) $$

Step-by-Step Solution

Answer

The simplified expression is \(-12x - 18\).

Step 1: Distribute Negatives and Constants

First, distribute the negative and constants in the expression. For \(-6(x-1)\), distribute \(-6\) to both \(x\) and \(-1\): \(-6x + 6\). For \(-3(x+8)\), distribute \(-3\) to both \(x\) and \(8\): \(-3x - 24\). Now the expression becomes: \(-6x + 6 - 3x - 3x - 24\).

Step 2: Combine Like Terms

Combine all the \(x\) terms together and the constant terms together. The terms are:\(-6x, -3x, -3x\) and the constants are \(6, -24\). This gives:\((-6x - 3x - 3x) + (6 - 24)\).

Step 3: Simplify the Expression

Combine the \(x\) terms \((-6x - 3x - 3x) = -12x\). Combine the constant terms \((6 - 24 = -18)\). Therefore, the expression simplifies to \(-12x - 18\).

Key Concepts

Distribution in Algebra

In elementary algebra, distribution refers to spreading out or distributing a number across terms inside parentheses. This usually happens when you have a coefficient (a number in front of a bracket) that needs to be multiplied by each term inside the brackets.
In the given expression, \[-6(x-1)-3x-3(x+8)\]you have two instances where distribution is applied. First, you need to distribute -6 across \((x-1)\). The rule says to multiply -6 by each term inside the parentheses:

• Multiply -6 by x, getting -6x
• Multiply -6 by -1, getting 6

This changes \(-6(x-1)\) to \(-6x + 6\).
Next, you apply distribution again to distribute -3 across \((x+8)\):

• Multiply -3 by x, giving -3x
• Multiply -3 by 8, giving -24

Thus, \(-3(x+8)\) becomes \(-3x - 24\).
It's important to apply this process correctly to maintain the original mathematical relationships in the expression.

Combining Like Terms

Combining like terms is a method to simplify expressions in algebra by adding or subtracting terms that have the exact same variable component.
For example, in the expression \[-6x + 6 - 3x - 3x - 24\],there are similar terms that can be combined:

• The terms \(-6x, -3x,\) and \(-3x\) are all like terms. They all involve x. Add them together: \(-6x - 3x - 3x = -12x\).
• The constants \(6\) and \(-24\) are also like terms. Combine them by performing the arithmetic: \(6 - 24 = -18\).

By effectively gathering these like terms, you reduce the complexity of the expression, making it easier to handle in further steps such as solving or simplifying.

Simplifying Expressions

Simplifying an expression in algebra involves combining all like terms that you've distributed and arranged. After performing distribution and combining like terms, you're left with a much simpler form to handle.
Take, for instance, the expression we derived earlier:\(-12x - 18\).Simplifying is crucial because it expresses the problem as efficiently as possible, making calculations or further transformations easier.
By simplifying expressions through the proper application of algebraic principles like distribution and combining like terms, complex algebra problems become much easier to manage and solve. It's essential for solving equations and performing accurate operations in algebra, and it forms a fundamental part of elementary algebra education that every student must understand.