Problem 18 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-12+(7-8 x)+6\)

Step-by-Step Solution

Answer

1 - 8x

Step 1 - Solve Inside the Parentheses

First, simplify the expression inside the parentheses. Here, we have \(7 - 8x\). Since we cannot combine a constant and a variable term, this expression remains as \(7 - 8x\).

Step 2 - Substitute Simplified Expression Back

Substitute the simplified expression from the parentheses back into the original equation. This gives us: \(-12 + (7 - 8x) + 6\).

Step 3 - Remove Parentheses and Combine Like Terms

Remove the parentheses and combine like terms. The expression now is: \(-12 + 7 - 8x + 6\). Combine the constant terms: \(-12 + 7 + 6\). This simplifies to \(1\). So, the expression is now: \(1 - 8x\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental concept in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. For example, in the expression `2x + 3x`, both terms are like terms because they both have the variable `x`.
To combine them, you add or subtract their coefficients. In our exercise, we combined `-12`, `7`, and `6`, which are all constants (terms without variables). This led to `-12 + 7 + 6`, simplifying to `1`.
Here’s another example for clarity: If you have `4y + 5y - y`, you combine like terms to get `8y` because `(4 + 5 - 1)y` simplifies to `8y`.
Remember, combining like terms helps streamline algebraic expressions into simpler forms, making them easier to solve or further manipulate.

Parentheses

Parentheses are used in algebraic expressions to indicate which operations should be performed first. They act as a grouping tool.
In our exercise, we started by simplifying the expression inside the parentheses `7 - 8x`. Parentheses ensure the proper order of operations, following the rule of PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Once you've simplified the contents inside the parentheses, you can remove them and proceed with the rest of the expression. Here, after simplifying `7 - 8x`, we placed it back into the expression, then removed the parentheses and combined like terms.

Variable Terms

Variable terms are parts of algebraic expressions that consist of variables (e.g., `x`, `y`, `z`) along with coefficients. They are different from constants, which are fixed values without variables.
In our exercise, `-8x` is the variable term. We couldn't combine it with constants (`-12`, `7`, and `6`) because they are not like terms. It's important to distinguish between variable terms and constants to simplify an expression correctly.
For instance, in an expression like `3a + 5 - 2a + 4`, we combine the variable terms `3a` and `-2a` to get `a`, and combine the constants `5` and `4` to get `9`. The simplified expression would be `a + 9`.
Understanding variable terms allows you to accurately manipulate and simplify algebraic expressions.